Problems – Limit of Algebraic Functions

LIMIT OF ALGEBRAIC FUNCTIONS — Practice Questions

30 problems on limits of algebraic functions.
Try to solve each question independently before checking the answer key.

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Instructions:
Choose the best answer. Correct: +4, Wrong: −1, Blank: 0.

Quick Notes — Limit Rules:
\(\displaystyle \lim_{x \to a} k = k\)
\(\displaystyle \lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)\)
\(\displaystyle \lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)\)
\(\displaystyle \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\)
\(\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}\), for \(\lim_{x \to a} g(x) \neq 0\)
\(\displaystyle \lim_{x \to a} [f(x)]^n = \left[\lim_{x \to a} f(x)\right]^n\)
\(\displaystyle \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}\)

Problem 1

Consider the following graph for problems 1 and 2.

Find \(\displaystyle \lim_{x \to 1} f(x) = \cdots \cdot\)

A. \(1\) B. \(2\) C. \(3\) D. \(5\) E. \(\text{undefined}\)

Problem 2

Using the same graph, find \(\displaystyle \lim_{x \to 3} f(x) = \cdots \cdot\)

A. \(0\) B. \(3\) C. \(5\) D. \(8\) E. \(\text{undefined}\)

Problem 3

Given \[ f(x) = \begin{cases} 2x+1, & \text{for } x < 3 \\[4pt] 3x, & \text{for } x \geq 3. \end{cases} \]

Find \(\displaystyle \lim_{x \to 1} f(x) = \cdots \cdot\)

A. \(-2\) B. \(-1\) C. \(1\) D. \(2\) E. \(3\)

Problem 4

Find \(\displaystyle \lim_{x \to -3} 2x + \lim_{x \to -2} (x^2-5)^3 = \cdots \cdot\)

A. \(-9\) B. \(-7\) C. \(-5\) D. \(-3\) E. \(-1\)

Problem 5

Find \[\displaystyle \lim_{x \to 4} \sqrt[3]{3x^2+7x-12} + \lim_{x \to 5} \bigl(\sqrt{3x^2-11}-3x\bigr) = \cdots \cdot\]

A. \(-7\) B. \(-4\) C. \(-3\) D. \(3\) E. \(4\)

Problem 6

Given \(\displaystyle \lim_{x \to 5} f(x)=2\) and \(\displaystyle \lim_{x \to 5} g(x)=-1\). Find \(\displaystyle \lim_{x \to 5} (f^2(x)-g^2(x)) = \cdots \cdot\)

A. \(-5\) B. \(-3\) C. \(2\) D. \(3\) E. \(5\)

Problem 7

Given \(\displaystyle \lim_{x \to 2} (2x^2-px+5) = -1\). Find the value of \(p\).

A. \(-7\) B. \(-6\) C. \(-2\) D. \(2\) E. \(7\)

Problem 8

Given \(f(x)=3-4x\). If \(\displaystyle \lim_{x \to p} f(x)=p-2\), find the value of \(p\).

A. \(1\) B. \(\dfrac35\) C. \(-1\) D. \(-\dfrac53\) E. \(2\)

Problem 9

Given \(\displaystyle \lim_{x \to a} f(x) = m\). If \(f(x)=2x\), find \(\displaystyle \lim_{x \to a} f(x^2-1) = \cdots \cdot\)

A. \(\dfrac12m^2-1\) B. \(\dfrac12m^2-2\) C. \(m^2-1\) D. \(m^2-2\) E. \(2m^2-1\)

Problem 10

A car moves with instantaneous velocity \(v(t)=t^2-t\) (in m/s). Find the velocity of the car as \(t\) approaches \(5\) seconds.

A. \(10\) m/s B. \(12\) m/s C. \(15\) m/s D. \(20\) m/s E. \(25\) m/s

Problem 11

The population growth rate is given by \(p(t) = \sqrt{\dfrac12t^2-3t+5}\) (in percent per year). Find the growth rate as \(t\) approaches \(5\) years.

A. \(0.75\%\) B. \(\sqrt{1.5}\%\) C. \(\sqrt{2}\%\) D. \(\sqrt{2.5}\%\) E. \(\sqrt{2.75}\%\)

Problem 12

Find \(\displaystyle \lim_{x \to 1} \dfrac{x^2-1}{x-1} = \cdots \cdot\)

A. \(-2\) B. \(-1\) C. \(0\) D. \(1\) E. \(2\)

Problem 13

Find \(\displaystyle \lim_{x \to 2} \dfrac{x^2+x-6}{x^2-4} = \cdots \cdot\)

A. \(-\dfrac54\) B. \(-\dfrac45\) C. \(\dfrac23\) D. \(\dfrac45\) E. \(\dfrac54\)

Problem 14

Find \(\displaystyle \lim_{x \to 4} \dfrac{3x^3-48x}{x^2-16} = \cdots \cdot\)

A. \(4\) B. \(12\) C. \(16\) D. \(24\) E. \(48\)

Problem 15

Find \(\displaystyle \lim_{x \to 2} \left(\dfrac{2}{x-2}-\dfrac{8}{x^2-4}\right) = \cdots \cdot\)

A. \(-\dfrac14\) B. \(-\dfrac12\) C. \(0\) D. \(\dfrac14\) E. \(\dfrac12\)

Problem 16

Find \(\displaystyle \lim_{x \to 2} \left(\dfrac{6}{x^2-x-2}-\dfrac{2}{x-2}\right) = \cdots \cdot\)

A. \(-1\) B. \(-\dfrac23\) C. \(-\dfrac13\) D. \(\dfrac13\) E. \(\dfrac23\)

Problem 17

Find \(\displaystyle \lim_{x \to 27} \dfrac{x-27}{x^{\frac{1}{3}}-3} = \cdots \cdot\)

A. \(27\) B. \(18\) C. \(9\) D. \(3\) E. \(1\)

Problem 18

Find \(\displaystyle \lim_{x \to 1} \dfrac{1-\sqrt{x}}{1-x^2} = \cdots \cdot\)

A. \(0\) B. \(\dfrac12\) C. \(\dfrac14\) D. \(1\) E. \(4\)

Problem 19

Find \(\displaystyle \lim_{x \to 0} \dfrac{x+\sqrt{x}}{\sqrt{x}} = \cdots \cdot\)

A. \(0\) B. \(\sqrt2\) C. \(1\) D. \(2\) E. \(3\)

Problem 20

Find \(\displaystyle \lim_{x \to 0} \dfrac{5x}{3-\sqrt{9+x}} = \cdots \cdot\)

A. \(-30\) B. \(-10\) C. \(10\) D. \(30\) E. \(50\)

Problem 21

Find \(\displaystyle \lim_{x \to 3} \dfrac{2-\sqrt{x+1}}{x-3} = \cdots \cdot\)

A. \(-\dfrac12\) B. \(-\dfrac14\) C. \(0\) D. \(\dfrac14\) E. \(\dfrac12\)

Problem 22

Find \(\displaystyle \lim_{x \to 2} \dfrac{4-x^2}{3-\sqrt{x^2+5}} = \cdots \cdot\)

A. \(2\) B. \(4\) C. \(6\) D. \(8\) E. \(10\)

Problem 23

Find \(\displaystyle \lim_{x \to 4} \dfrac{x-4}{\sqrt{x}-2} = \cdots \cdot\)

A. \(0\) B. \(2\) C. \(4\) D. \(8\) E. \(10\)

Problem 24

Find \(\displaystyle \lim_{x \to \sqrt{2}} \dfrac{x^2-2}{x-\sqrt{2}} = \cdots \cdot\)

A. \(-2\sqrt2\) B. \(-\sqrt2\) C. \(0\) D. \(\sqrt2\) E. \(2\sqrt2\)

Problem 25

Find \(\displaystyle \lim_{x \to 3} \dfrac{\sqrt{2x-2}-2}{\sqrt{3x}-3} = \cdots \cdot\)

A. \(0\) B. \(\dfrac23\) C. \(\dfrac23\sqrt3\) D. \(1\) E. \(\dfrac32\)

Problem 26

Find \(\displaystyle \lim_{x \to 3} \dfrac{\sqrt{x+4}-\sqrt{2x+1}}{x-3} = \cdots \cdot\)

A. \(-\dfrac17\sqrt7\) B. \(-\dfrac{1}{14}\sqrt7\) C. \(0\) D. \(\dfrac17\sqrt7\) E. \(\dfrac{1}{14}\sqrt7\)

Problem 27

Find \(\displaystyle \lim_{x \to 2} \dfrac{\sqrt{3x^2+8x-3}-\sqrt{4x^2+9}}{x-2} = \cdots \cdot\)

A. \(-\dfrac45\) B. \(0\) C. \(\dfrac25\) D. \(\dfrac52\) E. \(\infty\)

Problem 28

If \(|f(x)-2| \leq x+3\), find \(\displaystyle \lim_{x \to -3} f(x) = \cdots \cdot\)

A. \(-2\) B. \(0\) C. \(1\) D. \(2\) E. \(3\)

Problem 29

If \(f(x) = \dfrac{x^2}{|x|} + 1\), find \(\displaystyle \lim_{x \to 0} f(x) + \lim_{x \to 1} f(x) = \cdots \cdot\)

A. \(0\) B. \(1\) C. \(3\) D. \(4\) E. \(5\)

Problem 30

Find \(\displaystyle \lim_{x \to 0} \dfrac{\sqrt[4]{1+x^4}-\sqrt{1+x^2}}{x^2} = \cdots \cdot\)

A. \(-1\) B. \(-\dfrac12\) C. \(0\) D. \(\dfrac12\) E. \(1\)