Before the midterm

Confusing the limit with the value of the function

The habit: A limit is about nearby points; the value at the point itself may differ or not exist.

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

Practise against it: 6 questions

What it looks like

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

Introductory1 mark · 2 min

A function $f$ is continuous at $x = 2$ , and $\lim_{x \to 2} f(x) = 5$ . Which of the following must be true?

With the slip: $f(x) = 5$ for every $x$ close to 2
Worked through: $\displaystyle f(2) = 5$

The limit says f(x) gets close to 5 as x approaches 2, not that it equals 5 there. f(x) = x + 3 is continuous with limit 5 at 2, yet f(1.9) = 4.9.

See the worked solution

Where it comes up

Limits1 question
Continuity4 questions
The definition of the derivative4 questions
Differentiation rules1 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
The function $f$ is defined piecewise below. Which statement about its behaviour at $x = 1$ is true?
LimitsIntroductory
02
The graph of a function $f$ is shown. Which statement about $f$ at $x = 2$ is true?
ContinuityIntroductory
03
The graph of a function $f$ is shown. Only the value $f(-1)$ may be changed; every other value stays as it is. What must $f(-1)$ be changed to so that $f$ is continuous at $x = -1$ ?
ContinuityIntroductory
04
A roastery’s cost of roasting $q$ kilograms of coffee in a day is $C(q)$ dollars, and $C'(50) = 12$ . Which statement is the best reading of $C'(50) = 12$ ?
The definition of the derivativeIntroductory
05
The graph of a function $g$ is shown. $g$ is continuous at $x = a$ when three conditions hold: first, $g(a)$ is defined; second, $\lim_{x \to a} g(x)$ exists; third, the two are equal. At each of $x = -2$ and $x = 1$ , which condition is the first to fail?
ContinuityStandard
06
For a base $a > 0$ , the slope of $y = a^{x}$ at $x = 0$ is the limit below. $\displaystyle \lim_{h \to 0} \frac{a^{h} - 1}{h}$
The definition of the derivativeStandard

Start the set