Limit at Infinity - Practice Questions

LIMIT AT INFINITY — Practice Questions

37 multiple-choice questions on limits at infinity.
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.

Problem 1

Find \(\displaystyle \lim_{x \to \infty} \dfrac{2x^3+3x^2-5x+4}{2x^4-4x^2+9} = \cdots \cdot\)

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

Problem 2

Find \(\displaystyle \lim_{x \to \infty} \dfrac{2x^3+3x^2+7}{x^2+3x+4} = \cdots \cdot\)

A. \(\infty\) B. \(0\) C. \(-\infty\) D. \(2\) E. \(\frac12\)

Problem 3

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

A. \(-4\) B. \(-3\) C. \(-2\) D. \(0\) E. \(\infty\)

Problem 4

If \(f(x) = x + \dfrac{x^2}{\sqrt{x^2-2x}}\), find \(\displaystyle \lim_{x \to \infty} \dfrac{f(x)}{x} = \cdots \cdot\)

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

Problem 5

Find \(\displaystyle \lim_{x \to \infty} \dfrac{\sqrt{18x^2-x+1}-3x}{\sqrt{x^2+2x}} = \cdots \cdot\)

A. \(\sqrt{15}\) B. \(3(\sqrt{2}-1)\) C. \(3(\sqrt{2}+1)\) D. \(3\) E. \(4.5\)

Problem 6

Find \(\displaystyle \lim_{x \to \infty} (\sqrt{x-\sqrt{x}} - \sqrt{x+\sqrt{x}}) = \cdots \cdot\)

A. \(0.5\) B. \(1\) C. \(-1\) D. \(0\) E. undefined

Problem 7

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

A. \(-\frac14\) B. \(-\frac12\) C. \(0\) D. \(\frac14\) E. \(\frac12\)

Problem 8

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

A. \(0\) B. \(\frac12\) C. \(1\) D. \(\frac32\) E. \(\frac52\)

Problem 9

Find \(\displaystyle \lim_{x \to \infty} (\sqrt{9x^2+5x+5} - \sqrt{9x^2-7x-4}) = \cdots \cdot\)

A. \(0\) B. \(\frac13\) C. \(1\) D. \(2\) E. \(3\)

Problem 10

Find \(\displaystyle \lim_{x \to \infty} (3x+1-\sqrt{9x^2+4x-7}) = \cdots \cdot\)

A. \(9\) B. \(6\) C. \(3\) D. \(\frac13\) E. \(\frac19\)

Problem 11

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

A. \(5\) B. \(3.5\) C. \(2.5\) D. \(1.5\) E. \(1\)

Problem 12

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

A. \(3\sqrt{2}-4\) B. \(\frac34\sqrt{2}-1\) C. \(\frac34-\sqrt{2}\) D. \(3-2\sqrt{2}\) E. \(\sqrt{2}-1\)

Problem 13

Find \(\displaystyle \lim_{x \to \infty} [\sqrt{(x+a)(x+b)} - x] = \cdots \cdot\)

A. \(\frac{a+b}{2}\) B. \(\frac{a-b}{2}\) C. \(\frac{ab}{2}\) D. \(\frac{2}{a+b}\) E. \(\frac{a+b}{a-b}\)

Problem 14

Find \(\displaystyle \lim_{x \to \infty} \sqrt{5(x-1)+2\sqrt{4x^2-23x-6}} = \cdots \cdot\)

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

Problem 15

Find \(\displaystyle \lim_{x \to \infty} 2x\left(\sqrt{9+\frac{10}{x}} - 3\right) = \cdots \cdot\)

A. \(\frac{10}{3}\) B. \(-\frac{10}{3}\) C. \(\frac53\) D. \(\frac53\sqrt{2}\) E. \(-\frac53\sqrt{2}\)

Problem 16

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

A. \(-\infty\) B. \(-\frac12\sqrt{2}\) C. \(0\) D. \(\frac12\sqrt{2}\) E. \(1\)

Problem 17

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

A. \(1\) B. \(\frac12\) C. \(\frac13\) D. \(\frac14\) E. \(\frac16\)

Problem 18

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

A. \(-2\) B. \(-1\) C. \(-\frac12\) D. \(\frac12\) E. \(\frac14\)

Problem 19

Find \(\displaystyle \lim_{x \to \infty} (\sqrt[3]{8x^3+12x^2-5} - \sqrt{x^2+6x} - x + 3) = \cdots \cdot\)

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

Problem 20

Find \(\displaystyle \lim_{n \to \infty} \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)\cdots\left(1-\frac{1}{n^2}\right) = \cdots \cdot\)

A. \(\infty\) B. \(2\) C. \(1\) D. \(\frac12\) E. \(\frac14\)

Problem 21

Find \(\displaystyle \lim_{x \to \infty} \dfrac{\sqrt{9x^2+12x-1}-\sqrt{9x^2-24x+10}}{\sqrt[3]{x^3+8x^2+x}-\sqrt[3]{x^3-x^2+2x}} = \cdots \cdot\)

A. \(\frac14\) B. \(\frac12\) C. \(1\) D. \(2\) E. \(3\)

Problem 22

Find \(\displaystyle \lim_{x \to \infty} \left(3x + \sin \frac{1}{x}\right) = \cdots \cdot\)

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

Problem 23

Find \(\displaystyle \lim_{x \to \infty} \left(2 + \cos \frac{4}{x}\right) = \cdots \cdot\)

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

Problem 24

Find \(\displaystyle \lim_{x \to \infty} \left(\tan \frac{1}{x} - x\right) = \cdots \cdot\)

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

Problem 25

Find \(\displaystyle \lim_{x \to \infty} \sin \left(\frac{1}{x} - \frac{4\pi}{3}\right) = \cdots \cdot\)

A. \(-\infty\) B. \(0\) C. \(\frac12\sqrt{2}\) D. \(\frac12\sqrt{3}\) E. \(1\)

Problem 26

Find \(\displaystyle \lim_{x \to \infty} \left(\sin \left(\frac{1}{x} - \frac{6\pi}{7}\right) - 5x\right) = \cdots \cdot\)

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

Problem 27

Find \(\displaystyle \lim_{y \to \infty} \sqrt{6y} \cos \frac{3}{\sqrt{y}} \sin \frac{5}{\sqrt{y}} = \cdots \cdot\)

A. \(0\) B. \(\sqrt{6}\) C. \(2\sqrt{6}\) D. \(5\sqrt{6}\) E. \(\infty\)

Problem 28

Find \(\displaystyle \lim_{x \to \infty} \dfrac{1-\cos \frac{4}{x}}{\frac{1}{x} \cdot \tan \frac{3}{x}} = \cdots \cdot\)

A. \(\frac13\) B. \(\frac23\) C. \(\frac43\) D. \(\frac83\) E. \(\infty\)

Problem 29

Find \(\displaystyle \lim_{x \to \infty} 2x^2\left(1-\cos \frac{6}{x}\right) = \cdots \cdot\)

A. \(0\) B. \(6\) C. \(12\) D. \(18\) E. \(36\)

Problem 30

Find \(\displaystyle \lim_{\theta \to \infty} \dfrac{\sin^2 \theta}{\theta^2-5} = \cdots \cdot\)

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

Problem 31

Find \(\displaystyle \lim_{y \to \infty} y \cdot \sin \frac{3}{y} \cdot \cos \frac{5}{y} = \cdots \cdot\)

A. \(0\) B. \(1\) C. \(2\) D. \(3\) E. \(4\)

Problem 32

Find \(\displaystyle \lim_{x \to \infty} \dfrac{\sin \frac{3}{x}}{\left(1-\cos \frac{2}{x}\right) \cdot x^2 \cdot \sin \frac{1}{x}} = \cdots \cdot\)

A. \(0\) B. \(\frac23\) C. \(1\) D. \(\frac32\) E. \(3\)

Problem 33

Find \(\displaystyle \lim_{x \to \infty} x\left(1-\cos \frac{1}{\sqrt{x}}\right) = \cdots \cdot\)

A. \(1\) B. \(\frac12\) C. \(\frac13\) D. \(\frac14\) E. \(\frac15\)

Problem 34

Find \(\displaystyle \lim_{x \to \infty} 2x \cdot \tan \frac{1}{x} \cdot \sec \frac{2}{x} = \cdots \cdot\)

A. \(0\) B. \(1\) C. \(2\) D. \(3\) E. \(4\)

Problem 35

Find \(\displaystyle \lim_{x \to \infty} \dfrac{2x^2 \tan \left(\frac{1}{x}\right) - x \sin \left(\frac{1}{x}\right) + \frac{1}{x}}{x \cos \left(\frac{2}{x}\right)} = \cdots \cdot\)

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

Problem 36

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

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

Problem 37

If \(a_n = \dfrac{n-2}{n}\) and \(a_n \to \ell\), find the smallest positive integer \(k\) such that \(|a_k-\ell| < \frac{1}{100}\).

A. \(100\) B. \(101\) C. \(199\) D. \(200\) E. \(201\)