Precalculus Introduction to Limits Test Solutions and Scoring (PDF)

Precalculus math test solutions: Introduction to Limits – Maths PDF

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This limits test answers for Precalculus covers introduction to limits (Common Core F-IF.B.6, F-IF.C.7d), with a free PDF to print.

Worked solutions and scoring notes for the 9-question limits test. Use the scoring line under each solution to award partial credit.

Precalculus · Common Core State Standards: F-IF.B.6, F-IF.C.7d

Limits test answers: worked solutions

Question 1 – Average rate of change [2 pts]

$f(4)=4$ and $f(1)=-2$. $\frac{4-(-2)}{3}=2$.

Scoring: 1 point for the two values, 1 point for the quotient.

Question 2 – Limit from a table [2 pts]

The values approach $2$ from both sides. At $x=1$ the denominator is zero.

The limit is 2

Scoring: 1 point for the limit, 1 point for the explanation.

Question 3 – Factor and cancel [2 pts]

$\frac{(x-5)(x+5)}{x-5}=x+5$.

Scoring: 1 point for factoring, 1 point for the limit.

Question 4 – Rationalize [2 pts]

$\frac{(\sqrt x-3)(\sqrt x+3)}{(x-9)(\sqrt x+3)}=\frac{1}{\sqrt x+3}\to\frac16$.

$\frac16$

Scoring: 1 point for rationalizing, 1 point for the limit.

Question 5 – Read a graph [3 pts]

Both pieces approach height 3.

(a) Both are $3$. (b) The limit is $3$, and $f(1)=1$. (c) No, since $f(1)\neq\lim f(x)$; it is a removable discontinuity.

$3$ and $3$; limit $3$, $f(1)=1$; not continuous

Scoring: 1 point for each part.

Question 6 – Limits at infinity [2 pts]

(a) Equal degrees: $\frac62=3$. (b) The denominator has the higher degree: 0.

(a) $3$; (b) $0$

Scoring: 1 point each.

Question 7 – Infinite limits [2 pts]

Slightly above 3, $x-3$ is a small positive number. Slightly below, it is a small negative number.

$\infty$ and $-\infty$

Scoring: 1 point each.

Question 8 – Step function [3 pts]

At w = 2 the fee jumps from 8 to 11 dollars.

(a) $\$8$. (b) $\$11$. (c) No, because the one-sided limits are different.

$\$8$; $\$11$; no

Scoring: 1 point for each part.

Question 9 – Continuity [2 pts]

Left limit $3$. Right value $2k-3$. Set $2k-3=3$.

$k=3$

Scoring: 1 point for the equation, 1 point for k.

Total score

Total

20 points

Key facts behind this limits test answers

Key facts

Average rate of change: (f(b) – f(a))/(b – a), the secant slope.
lim x to a of f(x) = L means f(x) approaches L as x approaches a from both sides.
The limit can exist even if f(a) is undefined (a hole).
Two-sided limit exists only if the left and right limits are equal.
f is continuous at a if f(a) exists, the limit exists and they are equal.

How to use this limits test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All Precalculus chapters: Precalculus math.

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