MATH 203 Calculus I • Concordia University, Montreal

Revision sheet: rates of change and limit laws (MATH 203)

Q: What is the difference between the average rate of change and the slope of the tangent?

The average rate of change is the slope of a secant line through two points of the graph: change in output divided by change in input. The slope of the tangent is what those secant slopes approach as the second point slides toward the first. It is a limit, estimated with a table of secant slopes and found exactly by rewriting the quotient.

Q: How do I estimate the slope of a tangent line with a calculator?

Compute the slope of the secant from the point to a second point at distance h, for h equal to 0.1, 0.01 and 0.001, and also for the negative values. Keep all the digits until the final division. The two lists close in on the tangent slope from opposite sides, and the bracket tells you how many decimals your estimate deserves.

Q: What do I do when I plug in and get 0 over 0?

Do not stop and do not say the limit does not exist. Zero over zero means the top and bottom share a factor. Factor the polynomials, multiply by the conjugate if there is a square root, or combine fractions over a common denominator after rewriting any negative exponent. Cancel the common factor, allowed because x is not equal to c, then substitute again.

Q: Can I trust a table of values to find a limit?

A table suggests a limit, it never proves one. The calculator can round a difference of nearly equal numbers to zero, and badly chosen points can hide an oscillation, as with the cosine of pi over x at x equal to 0.1, 0.01 and 0.001. Use the table as a target, then decide the limit with algebra and the limit laws.

Q: When do I use the Sandwich Theorem?

Use it when one factor has no limit but stays bounded, typically the sine or cosine of one over x, and the rest tends to zero, or when the function is only given through inequalities. Build two bounds valid for every x near the point, check that they have the same limit, and name the theorem in your conclusion.

This sheet is not a summary of sections 2.1 and 2.2 of Thomas' Calculus: you have the textbook. It answers one question only, what makes students lose marks on rates of change and the limit laws in MATH 203 at Concordia University, and which precise gesture avoids each loss.

The limits of this chapter are rarely hard; the algebra around them is. A fraction inside a fraction, a negative exponent, a conjugate multiplied on one side only, a factorization that forgets its leading coefficient: these cost more marks than any limit law. Your scientific calculator helps with the tables of secant slopes, and every trap below says when it can be trusted and when it cannot.

Mark this sheet as read or add it to your favourites: a free account, no password, keeps your read sheets and favourites from one visit to the next and tells you which chapter to tackle next. Create your space, an email is enough.

The thread of the chapter

The secant is the only slope you can compute and the tangent is what the secants approach; reaching it means a limit whose substitution gives $\frac{0}{0}$ , an order to rewrite, and the marks are lost in the ALGEBRA of that rewriting, not in the limit.

This chapter is part of MATH 203, Calculus I (Concordia)

What is on this sheet

1. The essentials
2. The rules in table form
3. The mistakes that cost marks
4. Which method to choose
5. How the answer is expected to be written
6. Check before you hand in
7. The typical problem, taken apart
8. Learn by heart
9. Frequently asked questions

Topics covered

Average rate of change
Secant line
Tangent line
Instantaneous rate of change
Limit of a function
Limit laws
Limits of rational functions
Indeterminate form 0/0
Conjugate of a radical expression
Complex fraction
Sandwich Theorem

The essentials

Secant, tangent, and the limit between them

• Average rate of change of $f$ on $[a, b]$ : $\frac{f(b) - f(a)}{b - a}$ , OUTPUT change over INPUT change. It is the slope of the secant through $(a, f(a))$ and $(b, f(b))$ , with the units of $f$ per unit of $x$ .
• The slope of the tangent at $P$ is the LIMIT of the slopes of the secants $PQ$ as $Q$ slides to $P$ . With $Q$ at $P$ itself the slope is $\frac{0}{0}$ , which is why a limit is needed at all.
• Without the exact tool, estimate: secants on BOTH sides of $P$ , closer and closer. On a curve that bends, one side overestimates and the other underestimates, so the two lists bracket the tangent slope.
• The average velocity on $[a, a + h]$ is a secant slope; the velocity at the instant $a$ is its limit as $h \to 0$ .

From

P(4, 2)

y = \sqrt x

, the secant to

Q(9, 3)

has slope

\frac{1}{5}

; as

Q

slides toward

P

the secant slopes rise toward the tangent slope

\frac{1}{4}

On a calculator table, keep every digit until the last division: $2^{1.001} - 2 \approx 0.0014$ , and rounding $2^{1.001}$ to $2.001$ first destroys the estimate.

The limit laws, their permits, and the licence to cancel

• If $\lim f = L$ and $\lim g = M$ are FINITE, the limits of $f \pm g$ , $kf$ , $fg$ , $f^n$ are $L \pm M$ , $kL$ , $LM$ , $L^n$ . The quotient law needs one more permit: $M \ne 0$ .
• Polynomials, and rational functions whose denominator is not $0$ at $c$ : the limit is the value at $c$ (Thomas, Theorems 2 and 3 of 2.2). Everywhere else, substituting is not a method.
• If $f(x) = g(x)$ for every $x \ne c$ near $c$ , then $f$ and $g$ have the same limit at $c$ . This is the licence to cancel a factor $x - c$ after the form $\frac{0}{0}$ .
• The value $f(c)$ plays no part in $\lim_{x\to c} f(x)$ : the limit reads the graph AROUND $c$ .
• Sandwich Theorem: $g \le f \le h$ near $c$ and $\lim g = \lim h = L$ give $\lim f = L$ . The two bounds must share the SAME limit.

Name the law on the line where you use it, and check its permit first: a quotient law used on a denominator that tends to $0$ earns nothing, even with the right number.

The rules in table form

Each row reads from left to right: the assumptions, then the result. A red cell is not an answer, it is the finding that the form settles nothing and the instruction to rewrite it. Every case is followed by a worked example.

Algebra inside a limit: the rules, and the rules that do not exist

Read a line as: this expression may be rewritten this way, with this result. The red lines are not answers: the first two are rewrites that are FALSE, the last is a form that decides nothing until the expression is rewritten.

Expression	Rewrite	Result
$\frac{1}{b} - \frac{1}{a}$	common denominator	$\frac{a - b}{ab}$
Example: $\frac{1}{5} - \frac{1}{2} = \frac{2 - 5}{10} = -\frac{3}{10}$ , then divided by $3$ : the rate of $\frac{1}{x}$ on $[2, 5]$ is $-\frac{1}{10}$ .
$x^{-1}$ , $x^{-1/2}$	reciprocal	$\frac{1}{x}$ , $\frac{1}{\sqrt x}$
Example: $\frac{x^{-1} + \frac{1}{2}}{x + 2} = \frac{1}{2x} \to -\frac{1}{4}$ at $-2$ ; with $x^{-1} = -x$ the top does not even vanish.
$\sqrt A - B$	times $\sqrt A + B$	$A - B^2$
Example: $\frac{\sqrt{3x - 2} - 2}{x - 2} = \frac{3}{\sqrt{3x - 2} + 2} \to \frac{3}{4}$ at $2$ .
$\sqrt{a + b}$	$\sqrt a + \sqrt b$	no such rule no such rule
Example: $\sqrt{9 + 16} = 5$ while $\sqrt 9 + \sqrt{16} = 7$ . What to do: Keep the root whole; to get rid of it in a limit, multiply top and bottom by the conjugate.
$(a - b)^{-1}$	$a^{-1} - b^{-1}$	no such rule no such rule
Example: $(1 - 3)^{-1} = -\frac{1}{2}$ while $1^{-1} - 3^{-1} = \frac{2}{3}$ . What to do: Rewrite $a^{-1} - b^{-1} = \frac{b - a}{ab}$ over a common denominator.
$\frac{0}{0}$ at $c$	substitution	indeterminate settles nothing
Example: $\frac{x^2 - 2x - 8}{x^2 - 16} = \frac{x + 2}{x + 4} \to \frac{3}{4}$ at $4$ . Same form, other result: At $1$ , $\frac{x^{10} - 1}{x - 1} \to 10$ while $\frac{(x - 1)^2}{x - 1} \to 0$ : same form, two answers. What to do: Factor $x - c$ out of top and bottom (or conjugate, or common denominator), cancel it for $x \ne c$ , substitute again.

Every false rewrite in this table turns a limit that exists into one that seems not to, or the reverse. Test any rewrite at one simple value, such as $x = 1$ or $x = 9$ , before building on it.

The mistakes that cost marks

These are the errors I correct most often in session. Each one costs marks on a paper, even when the reasoning behind it is right.

1. Computing a rate upside down

the whole part, and the unit mark

What not to write

“The average rate of change of $f$ on $[-1, 0]$ is $\frac{0 - (-1)}{f(0) - f(-1)} = \frac{1}{4}$ .”

What to write

“ $\frac{f(0) - f(-1)}{0 - (-1)} = \frac{5 - 1}{1} = 4$ .”

Why: A rate is output change per unit of input: $\frac{\Delta y}{\Delta x}$ . The units say it too: cells per HOUR, metres per SECOND.

2. Estimating a tangent slope from one side only

1 to 2 marks: the estimate is off in the first decimal

What not to write

“With $h = 0.1$ , the slope of $y = 2^x$ at $P(1, 2)$ is $1.4355$ .”

What to write

“Right side: $1.4355, 1.3911, 1.3868$ ; left side: $1.3393, 1.3815, 1.3858$ . The slope is between $1.3858$ and $1.3868$ , about $1.39$ .”

Why: A curve that bends upward makes every right-hand secant too steep and every left-hand one too flat. Two sides give a bracket, one side gives a biased guess.

3. Reading 0/0 as a verdict

the whole limit

What not to write

“Substitution gives $\frac{0}{0}$ , so $\lim_{x\to 2} \frac{x^2 + x - 6}{x - 2}$ does not exist.”

What to write

“The form is $\frac{0}{0}$ . For $x \ne 2$ , $\frac{(x - 2)(x + 3)}{x - 2} = x + 3$ , so the limit is $5$ .”

\frac{x^2 + x - 6}{x - 2}

is the line

y = x + 3

with the point

(2, 5)

removed: undefined at

2

, and yet its limit there is

5

Why: The zeros on top and bottom say that $x - 2$ is a common factor. After cancelling, the function is a line with a HOLE, and the limit is the height of the hole.

4. Turning a negative exponent into a minus sign

the whole question

What not to write

“ $x^{-1} + \frac{1}{2} = -x + \frac{1}{2}$ , which is $\frac{5}{2}$ at $-2$ , so the limit of $\frac{x^{-1} + \frac{1}{2}}{x + 2}$ does not exist.”

What to write

“ $x^{-1} + \frac{1}{2} = \frac{1}{x} + \frac{1}{2} = \frac{x + 2}{2x}$ , so the quotient is $\frac{1}{2x}$ for $x \ne -2$ , and the limit is $-\frac{1}{4}$ .”

Why: $x^{-n} = \frac{1}{x^n}$ : a negative exponent means a reciprocal. Rewrite every negative or fractional exponent BEFORE doing anything else.

5. Splitting the square root of a sum

the whole question

What not to write

“ $\sqrt{x + 16} = \sqrt x + 4$ , so $\frac{\sqrt{x + 16} - 4}{x} = \frac{1}{\sqrt x}$ .”

What to write

“With the conjugate, $\frac{\sqrt{x + 16} - 4}{x} = \frac{1}{\sqrt{x + 16} + 4}$ for $x \ne 0$ , so the limit at $0$ is $\frac{1}{8}$ .”

Why: The root of a sum is not the sum of the roots: $\sqrt{25} = 5$ , $\sqrt 9 + 4 = 7$ . Only the conjugate removes a root legally.

6. Losing the leading coefficient in a factorization

2 marks, although the method is right

What not to write

“ $2x^2 + 5x - 3 = (x + 3)\left(x - \frac{1}{2}\right)$ , so the limit at $-3$ of $\frac{2x^2 + 5x - 3}{x^2 + 3x}$ is $\frac{7}{6}$ .”

What to write

“ $2x^2 + 5x - 3 = (2x - 1)(x + 3)$ , so the quotient is $\frac{2x - 1}{x}$ for $x \ne -3$ , and the limit is $\frac{7}{3}$ .”

Why: A quadratic with leading coefficient $a$ factors as $a(x - r_1)(x - r_2)$ . Expanding the product back takes ten seconds and catches the missing $2$ .

7. Using the quotient law when the denominator tends to 0

1 to 2 method marks

What not to write

“ $\lim_{x\to 1} \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)} = \frac{2}{0}$ .”

What to write

“ $\lim_{x\to 1} g(x) = 0$ , so the quotient law does not apply; the expression must be studied another way.”

Why: Every limit law has a permit: the limits of the pieces must exist, and for a quotient the limit of the denominator must not be $0$ . Write the permit before the law.

8. Trusting a table the calculator has rounded

the whole question

What not to write

“ $G(0.1) = G(0.01) = G(0.001) = 1$ for $G(x) = \cos\frac{\pi}{x}$ , so the limit at $0$ is $1$ .”

What to write

“ $G\left(\frac{1}{2k}\right) = 1$ but $G\left(\frac{1}{2k + 1}\right) = -1$ , arbitrarily close to $0$ : the limit does not exist.”

\cos\frac{\pi}{x}

swings between

1

(green dots,

x = \frac{1}{2k}

) and

-1

(red dots,

x = \frac{1}{2k + 1}

) faster and faster as

x \to 0

Why: A table only sees the points you chose; the points $10^{-n}$ all make $\frac{\pi}{x}$ an even multiple of $\pi$ . And a table that jumps to $0$ for tiny $x$ , as $\frac{\sqrt{x^2 + 100} - 10}{x^2}$ does, is round-off.

Which method to choose

Which gesture, by the FORM of the limit

Substitute c first, on the side. What you get picks the tool

If a number, and the function is a polynomial or a rational function whose denominator is not $0$ at $c$ → done: the limit is the value; name the theorem

Example: $\lim_{x\to -1} \frac{x^2 + 4}{x - 3} = \frac{5}{-4} = -\frac{5}{4}$
If $\frac{0}{0}$ with polynomials → factor $x - c$ out of both: roots, sum or difference of cubes, grouping, synthetic division

Example: $\frac{x^3 - 2x^2 - 4x + 3}{x - 3} = x^2 + x - 1 \to 11$ at $3$

expand the factorization back before cancelling
If $\frac{0}{0}$ with a square root → multiply top AND bottom by the conjugate

Example: $\frac{\sqrt{x + 1} - 2}{\sqrt{x + 6} - 3} \to \frac{3}{2}$ at $3$ (two conjugates)
If $\frac{0}{0}$ with fractions inside a fraction, or negative exponents → rewrite the exponents, common denominator, then multiply by the reciprocal

Example: $\frac{1}{x}\left(\frac{1}{3 - x} - \frac{1}{3}\right) = \frac{1}{3(3 - x)} \to \frac{1}{9}$ at $0$
If a bounded factor ( $\sin$ or $\cos$ of something wild) times a factor tending to $0$ , or $f$ known only by inequalities → Sandwich Theorem, with two bounds that share their limit

Example: $\frac{x^2}{3} \le \frac{x^2}{2 + \sin(1/x)} \le x^2$ , so the limit at $0$ is $0$
If a number over something tending to 0 → the quotient law has no permit; say so, the study of such limits comes with infinite limits

Example: $\frac{f(x)}{g(x)}$ at $1$ with $f \to 2$ and $g \to 0$

No branch fits, or the algebra stalls? Estimate with a table of values on both sides, to know what you are aiming for, then return to the algebra: a table suggests, only the rewriting proves.

How the answer is expected to be written

A marker ticks steps. Here they are in order, with the concluding sentence expected word for word.

Estimating the slope of a tangent from a table of secants

When to use it: The question says estimate, or the exact slope needs a rule not yet available (an exponential, a data table)

1 Write the slope of the secant $PQ$ as a formula in the abscissa of $Q$ , or in $h$ : $\frac{f(a + h) - f(a)}{h}$ .
2 Compute it for $h = 0.1$ , $0.01$ , $0.001$ AND $h = -0.1$ , $-0.01$ , $-0.001$ , keeping every digit until the last division.
3 Observe the two lists: one decreasing, one increasing, closing in on the same number.
4 Conclude with the bracket and the number of decimals it justifies.

Concluding sentence

“The slopes of the secants to the right of $P$ decrease to $1.3868$ , those to the left increase to $1.3858$ ; the slope of the tangent at $P$ lies between them, so it is $1.39$ to two decimals.”

The trap: A single value of $h$ , or a single side: the estimate is then biased and nothing on the page shows it.

Marking: Usually 1 mark for the table on each side and 1 for the justified estimate.

The Sandwich Theorem, written for full marks

When to use it: A factor with no limit ( $\sin\frac{1}{x}$ , $\cos\frac{1}{x}$ ) that stays bounded, or a function given only through inequalities

1 Start from the bound of the wild factor, valid for every $x \ne c$ : $-1 \le \sin\frac{1}{x} \le 1$ .
2 Build the inequality step by step, stating each operation: adding a constant keeps it, multiplying by a POSITIVE quantity keeps it, taking reciprocals of positive numbers REVERSES it.
3 Compute the limits of the two bounds with the limit laws, and check that they are EQUAL.
4 Name the theorem and conclude.

Concluding sentence

“For all $x \ne 0$ , $\frac{x^2}{3} \le \frac{x^2}{2 + \sin(1/x)} \le x^2$ . Since $\lim_{x\to 0} \frac{x^2}{3} = \lim_{x\to 0} x^2 = 0$ , the Sandwich Theorem gives $\lim_{x\to 0} \frac{x^2}{2 + \sin(1/x)} = 0$ .”

The trap: Writing $-\theta \le \sin\theta \le \theta$ for all $\theta$ : false for $\theta < 0$ , where the bounds swap. Use $|\theta|$ .

Marking: Typically 1 mark for the bounds, 1 for their common limit, 1 for naming the theorem.

Check before you hand in

Five minutes of checking recover more marks than one more problem started in a hurry.

Test a rewrite at one simple value

Before cancelling or building on an identity, put x = 1 or x = 9 into both sides. A false rule fails at once.

$(x + 3)\left(x - \frac{1}{2}\right)$ against $2x^2 + 5x - 3$ at $x = 1$ : $2$ against $4$ , the leading coefficient is missing.
A table on both sides, as a target

Evaluate the expression at c plus and minus 0.01 with the calculator. The exact answer must be close to both values.

For $\frac{\sqrt{3x - 2} - 2}{x - 2}$ : $x = 2.01$ gives $0.7486$ , $x = 1.99$ gives $0.7514$ , so $\frac{3}{4}$ is plausible and $\frac{1}{4}$ is not.
The sign of the answer

Take x just beside c and read the sign of every factor, or read whether the curve rises or falls. The limit or the slope cannot have the opposite sign.

$\frac{1}{x}$ decreases on $[2, 5]$ , so its average rate there is negative: $-\frac{1}{10}$ , never $\frac{1}{10}$ .
Radian mode

Any sine or cosine in a rate or a limit is in radians. Check the mode before the first keystroke.

The rate of $\sin x$ on $\left[0, \frac{\pi}{6}\right]$ is $\frac{3}{\pi} \approx 0.9549$ ; degree mode returns $0.0167$ .

The typical problem, taken apart

From secants to the tangent, with a fraction inside a fraction

Let $f(x) = \frac{1}{x + 1}$ and $P\left(1, \frac{1}{2}\right)$ . Estimate the slope of the tangent at $P$ with two secants, then find it exactly as the limit of the secant slopes, and write the equation of the tangent.

The secant from

P

Q\left(3, \frac{1}{4}\right)

has slope

-\frac{1}{8}

; the tangent at

P

is steeper, and the question is by how much.

Step 1

Secant slopes with the calculator: $Q$ at $1.1$ gives $\frac{\frac{1}{2.1} - \frac{1}{2}}{0.1} \approx -0.2381$ , $Q$ at $0.9$ gives $\frac{\frac{1}{1.9} - \frac{1}{2}}{-0.1} \approx -0.2632$ . The slope is about $-0.25$ .

Why

Two sides give a bracket and a target: the exact answer must land between $-0.2632$ and $-0.2381$ .

Step 2

The slope of $PQ$ for $Q\left(x, \frac{1}{x + 1}\right)$ is $m(x) = \frac{\frac{1}{x + 1} - \frac{1}{2}}{x - 1}$ , $x \ne 1$ . At $x = 1$ it gives $\frac{0}{0}$ .

Why

Writing the form shows the marker why a rewriting follows; the tangent slope is $\lim_{x\to 1} m(x)$ .

Step 3

Common denominator on top: $\frac{1}{x + 1} - \frac{1}{2} = \frac{2 - (x + 1)}{2(x + 1)} = \frac{1 - x}{2(x + 1)}$ .

Why

This is the step where the marks go: the parentheses around $x + 1$ carry the minus sign to both terms.

Step 4

Divide by $x - 1$ , and use $1 - x = -(x - 1)$ : $m(x) = \frac{-(x - 1)}{2(x + 1)(x - 1)} = -\frac{1}{2(x + 1)}$ for $x \ne 1$ .

Why

The factor $x - 1$ appears only after turning $1 - x$ around; cancelling it is legal because $x \ne 1$ .

Step 5

$-\frac{1}{2(x + 1)}$ is rational with denominator $4 \ne 0$ at $1$ , so $\lim_{x\to 1} m(x) = -\frac{1}{4}$ . Tangent: $y = \frac{1}{2} - \frac{1}{4}(x - 1)$ , that is $y = -\frac{1}{4}x + \frac{3}{4}$ .

Why

Naming the theorem on rational functions justifies the substitution that was forbidden two lines earlier.

Step 6

Check: $-0.2632 < -0.25 < -0.2381$ , inside the bracket of the first step.

Why

A lost minus sign would give $+\frac{1}{4}$ , outside the bracket, and the check catches it for free.

The conclusion, written out

“The slopes of the secants $PQ$ tend to $-\frac{1}{4}$ as $Q \to P$ , so the tangent at $P\left(1, \frac{1}{2}\right)$ has slope $-\frac{1}{4}$ and equation $y = -\frac{1}{4}x + \frac{3}{4}$ .”

The classic mistake on this problem: Writing $\frac{1}{x + 1} - \frac{1}{2} = \frac{1}{x - 1}$ , or $2 - x + 1$ without parentheses, which gives $\frac{3 - x}{2(x + 1)}$ and no common factor.

Learn by heart

• Rate of change $= \frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$ : output over input, with units.
• Tangent slope $=$ limit of secant slopes; estimate it with secants on BOTH sides.
• Every limit law needs its permit: finite limits for the pieces, and $\lim g \ne 0$ for a quotient.
• Substituting is a method only for polynomials and rational functions defined at $c$ .
• $\frac{0}{0}$ is an order to rewrite: factor, conjugate, common denominator; cancel for $x \ne c$ .
• $x^{-n} = \frac{1}{x^n}$ ; $\sqrt{a + b} \ne \sqrt a + \sqrt b$ ; $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .
• Sandwich: two bounds with the SAME limit; $-|\theta| \le \sin\theta \le |\theta|$ .

Frequently asked questions

What is the difference between the average rate of change and the slope of the tangent?

The average rate of change is the slope of a secant line through two points of the graph: change in output divided by change in input. The slope of the tangent is what those secant slopes approach as the second point slides toward the first. It is a limit, estimated with a table of secant slopes and found exactly by rewriting the quotient.

How do I estimate the slope of a tangent line with a calculator?

Compute the slope of the secant from the point to a second point at distance h, for h equal to 0.1, 0.01 and 0.001, and also for the negative values. Keep all the digits until the final division. The two lists close in on the tangent slope from opposite sides, and the bracket tells you how many decimals your estimate deserves.

What do I do when I plug in and get 0 over 0?

Do not stop and do not say the limit does not exist. Zero over zero means the top and bottom share a factor. Factor the polynomials, multiply by the conjugate if there is a square root, or combine fractions over a common denominator after rewriting any negative exponent. Cancel the common factor, allowed because x is not equal to c, then substitute again.

Can I trust a table of values to find a limit?

A table suggests a limit, it never proves one. The calculator can round a difference of nearly equal numbers to zero, and badly chosen points can hide an oscillation, as with the cosine of pi over x at x equal to 0.1, 0.01 and 0.001. Use the table as a target, then decide the limit with algebra and the limit laws.

When do I use the Sandwich Theorem?

Use it when one factor has no limit but stays bounded, typically the sine or cosine of one over x, and the rest tends to zero, or when the function is only given through inequalities. Build two bounds valid for every x near the point, check that they have the same limit, and name the theorem in your conclusion.

Practise it

Corrected exercises: Rates of change and limit laws, MATH 203 at Concordia

A method is proved on a paper, not on a sheet. The set for the same chapter takes each of these traps into a problem, with the solution written out step by step.

10 corrected exercises
100 points
150 minutes

Do the exercises

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© Ahmed Squalli Houssaini. Revision sheet published at www.letuteurscientifique.ca/en/fiches/math203-tangent-lines-limit-laws. Free for personal and classroom use; republishing it elsewhere requires written permission (legal notice).

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Revision sheet: rates of change and limit laws (MATH 203)

The essentials

Secant, tangent, and the limit between them

The limit laws, their permits, and the licence to cancel

The rules in table form

Algebra inside a limit: the rules, and the rules that do not exist

The mistakes that cost marks

1. Computing a rate upside down

2. Estimating a tangent slope from one side only

3. Reading 0/0 as a verdict

4. Turning a negative exponent into a minus sign

5. Splitting the square root of a sum

6. Losing the leading coefficient in a factorization

7. Using the quotient law when the denominator tends to 0

8. Trusting a table the calculator has rounded

Which method to choose

Which gesture, by the FORM of the limit

How the answer is expected to be written

Estimating the slope of a tangent from a table of secants

The Sandwich Theorem, written for full marks

Check before you hand in

The typical problem, taken apart

From secants to the tangent, with a fraction inside a fraction

Learn by heart

Frequently asked questions

What is the difference between the average rate of change and the slope of the tangent?

How do I estimate the slope of a tangent line with a calculator?

What do I do when I plug in and get 0 over 0?

Can I trust a table of values to find a limit?

When do I use the Sandwich Theorem?

Corrected exercises: Rates of change and limit laws, MATH 203 at Concordia

See also

Looking for a MATH 203 tutor in Montreal?