Before Quiz 2

Treating sin(ax)/(bx) as the standard limit 1

The habit: sin(u)/u tends to 1 as u → 0, so the argument of the sine and the denominator must be the same; rescale so that they are.

8 practice questions are built to catch it, in 4 topics.

Practise against it: 6 questions

What it looks like

A question from L’Hôpital’s rule, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

Evaluate the limit.

\lim_{x \to \infty} x\sin\!\left(\frac{3}{x}\right)

With the slip: $\displaystyle 1$
Worked through: $\displaystyle 3$

In sin(3/x)/(1/x) the angle is 3/x but the denominator is 1/x, so they do not match: rescale the denominator to 3/x first.

See the worked solution

Where it comes up

Limits2 questions
Continuity2 questions
Exponential and trigonometric functions1 question
L’Hôpital’s rule3 questions

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
Find the value of the constant $c$ that makes $f$ continuous at $x = 0$ .
ContinuityIntroductory
02
Evaluate the limit. $\displaystyle \lim_{x \to 0} \frac{\sin(3x)}{5x}$
LimitsStandard
03
Evaluate the limit. $\displaystyle \lim_{x \to 0} \frac{1 - \cos^{2}(3x)}{4x^{2}}$
LimitsStandard
04
For which value of the constant $k$ is the function below continuous at $x = 0$ ?
ContinuityStandard
05
Evaluate the limit. $\displaystyle \lim_{x \to 0} \frac{\sin(3x)}{\tan(4x)}$
Exponential and trigonometric functionsStandard
06
Evaluate $\displaystyle\lim_{x \to 0} \frac{x^{2}\sin(1/x)}{\sin x}$ , or decide that it does not exist.
L’Hôpital’s ruleStandard