Before the midterm

Stopping at 0/0

The habit: 0/0 is not an answer; it says the expression has not been simplified yet.

21 practice questions are built to catch it, in 7 topics.

See it fail in a figure: Not stopped by 0/0, in the limits lesson

Practise against it: 6 questions

What it looks like

A question from limits, with the answer this slip gives beside the right one.

Introductory1 mark · 2 min

Evaluate the limit.

\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}

With the slip: $\displaystyle 0$
Worked through: $\displaystyle 6$

x² − 9 and x − 3 both vanish at 3, so 0/0 says the algebra is unfinished, not that the answer is zero: cancel the factor x − 3 first.

See the worked solution

Where it comes up

Limits7 questions
Continuity2 questions
The definition of the derivative3 questions
Differentiation rules1 question
Exponential and trigonometric functions2 questions
L’Hôpital’s rule5 questions
Taylor polynomials1 question

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

01
Which statement about the limit of the function below as $x \to 3$ is correct? $\displaystyle f(x) = \frac{x^{2} - x - 6}{|x - 3|}$
LimitsIntroductory
02
Evaluate the limit. $\displaystyle \lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x}$
LimitsStandard
03
Evaluate the one-sided limit. $\displaystyle \lim_{x \to 3^{-}} \frac{|x - 3|}{x^{2} - 9}$
LimitsStandard
04
Evaluate the limit. $\displaystyle \lim_{x \to 0} \frac{1 - \cos^{2}(3x)}{4x^{2}}$
LimitsStandard
05
There is exactly one value of the constant $a$ for which the limit below exists. Find that value of $a$ , and then enter the value of the limit. $\displaystyle \lim_{x \to 3} \frac{x^{2} + ax - 12}{x^{2} - x - 6}$
LimitsChallenging
06
Evaluate the limit. $\displaystyle \lim_{x \to 2} \frac{\sqrt{5x + 6} - \sqrt{x + 14}}{x - 2}$
LimitsChallenging