Limits Questions

Evaluate the following limits algebraically.

1) Simple Substitution

a) lim⁡x→3(2x+5)\lim_{x \to 3}(2x + 5)

b) lim⁡x→−2(x2+4x+5)\lim_{x \to -2}(x^2 + 4x + 5)

c) lim⁡x→1x2+xx+1\lim_{x \to 1}\frac{x^2 + x}{x + 1}

Answer Key – 1) Simple Substitution

a) 11
b) 1
c) 1

2) Simplification Needed (Factoring / Rationalization)

a) lim⁡x→2x2−4x−2\lim_{x \to 2}\frac{x^2 - 4}{x - 2}

b) lim⁡x→1x3−1x−1\lim_{x \to 1}\frac{x^3 - 1}{x - 1}

c) lim⁡x→4x−2x−4\lim_{x \to 4}\frac{\sqrt{x} - 2}{x - 4}

d) lim⁡x→01+x−1x\lim_{x \to 0}\frac{\sqrt{1 + x} - 1}{x}

e) lim⁡x→8x3−2x−8\lim_{x \to 8}\frac{\sqrt[3]{x} - 2}{x - 8}

f) lim⁡x→∞3x2+x+12x2−5\lim_{x \to \infty}\frac{3x^2 + x + 1}{2x^2 - 5}

g) lim⁡x→∞5x3−2x+13x3+x2\lim_{x \to \infty}\frac{5x^3 - 2x + 1}{3x^3 + x^2}

Answer Key – 2) Simplification Needed

a) 4
b) 3
c) 14\frac{1}{4}
d) 12\frac{1}{2}
e) 112\frac{1}{12}
f) 32\frac{3}{2}
g) 53\frac{5}{3}

3) Trigonometric Limits

a) lim⁡x→0sin⁡xx\lim_{x \to 0}\frac{\sin x}{x}

b) lim⁡x→0tan⁡(5x)x\lim_{x \to 0}\frac{\tan(5x)}{x}

c) lim⁡x→01−cos⁡xx2\lim_{x \to 0}\frac{1 - \cos x}{x^2}

d) lim⁡x→0x2cos⁡(1x)\lim_{x \to 0}x^2 \cos\left(\frac{1}{x}\right)

e) lim⁡x→0sin⁡(3x)sin⁡(2x)\lim_{x \to 0}\frac{\sin(3x)}{\sin(2x)}

Answer Key – 3) Trigonometric Limits

a) 1
b) 5
c) 12\frac{1}{2}
d) 0
e) 32\frac{3}{2}

4) Piecewise / Multi-Definition Functions

a) lim⁡x→3−∣x−3∣x−3\lim_{x \to 3^-}\frac{|x - 3|}{x - 3}

b) lim⁡x→2{x2−4x+5if x<2x+1if x≥2\lim_{x \to 2} \begin{cases} x^2 - 4x + 5 & \text{if } x < 2 \\ x + 1 & \text{if } x \geq 2 \end{cases}

c) lim⁡x→1{2x−1if x<1x2if x≥1\lim_{x \to 1} \begin{cases} 2x - 1 & \text{if } x < 1 \\ x^2 & \text{if } x \geq 1 \end{cases}

Answer Key – 4) Piecewise / Multi-Definition Functions

a) −1-1
b) DNE
c) Does exist (left limit is 1, right is 1) → But both are 1, so limit exists and is 1

5) Limits Involving Euler’s Number ee

a) lim⁡x→∞(1+1x)x\lim_{x \to \infty}\left(1 + \frac{1}{x}\right)^x

b) lim⁡x→0(1+x)1/x\lim_{x \to 0}\left(1 + x\right)^{1/x}

c) lim⁡x→0ex−1x\lim_{x \to 0}\frac{e^x - 1}{x}

d) lim⁡x→01−e−xx\lim_{x \to 0}\frac{1 - e^{-x}}{x}

e) lim⁡x→0e2x−1x\lim_{x \to 0}\frac{e^{2x} - 1}{x}

Answer Key – 5) Euler’s Number $e$

a) ee
b) ee
c) 1
d) 1
e) 2

6) Advanced Problems

a) lim⁡x→0+ln⁡(1+x)x\lim_{x \to 0^+} \frac{\ln(1 + x)}{x}

b) lim⁡x→0+xln⁡x\lim_{x \to 0^+} x \ln x

c) lim⁡x→0(1x−1sin⁡x)\lim_{x \to 0} \left(\frac{1}{x} - \frac{1}{\sin x}\right)

d) lim⁡x→∞(ln⁡(x+1)−ln⁡x)\lim_{x \to \infty} \left(\ln(x + 1) - \ln x\right)

e) lim⁡x→0+(sin⁡xx)1/x\lim_{x \to 0^+} \left(\frac{\sin x}{x}\right)^{1/x}

f) lim⁡x→∞(1+2x)3x\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{3x}

g) lim⁡x→0ex−cos⁡xx\lim_{x \to 0} \frac{e^x - \cos x}{x}

h) lim⁡x→0ln⁡(1+x2)xtan⁡x\lim_{x \to 0} \frac{\ln(1 + x^2)}{x \tan x}

Answer Key – 6) Advanced Problems

a) 1
b) 0
c) 0
d) 0
e) 1
f) e6e^6
g) 1
h) 1