Limits of Functions: Problems with Solutions

By Denitsa Dimitrova (Bulgaria)
Problem 1
$\lim_{x\to3}\frac{x^2-3x}{x-3}$
Problem 2
$\lim_{x\to-1}(2x^3-4x^2+5)$
Problem 3
$\lim_{x\to0}\frac{6x^3-5x^2+4x}{2x}$
Problem 4
$\lim_{x\to2}\frac{5x-10}{x^2-x-2}$
Problem 5
$\lim_{x\to0}\frac{5x^2}{3x^3+2x^2}$
Problem 6
$\lim_{x\to1}\left(\frac{x}{2x-2}-\frac{2-x}{x^2-1}\right)$
Problem 7
$\lim_{x\to \infin}\frac{6x^3-x+5}{8x^3+3x-10}$
Problem 8
$\lim_{x\to 0}\frac{x^2-1}{x^2+x}$
Problem 9
$\lim_{x\to\infin}\frac{x+2}{x^2+4x+5}$
Problem 10
$\lim_{x\to\infin}\frac{8x^4+5x^3}{4x^3+2x^2-x}$

Problem 11
$\lim_{x\to\infin}\frac{3x^2-10x}{3x-7x^2}$
Problem 12
$\lim_{x\to5}\frac{x-5}{\sqrt{x-1}-2}$
Problem 13
$\lim_{x\to1}\frac{x^2-2x+1}{x^3-x^2-x+1}$
Problem 14
$\lim_{x\to2}\frac{10-3x-x^2}{x^4-9x^2+20}$
Problem 15
$\lim_{x\to7}\frac{\sqrt{x-3}-2}{x-7}$
Problem 16
$\lim_{x\to0}\frac{x}{\sin x}$
Problem 17
$\lim_{x\to0}\frac{\sin 5x}{x}$
Problem 18
$\lim_{x\to 0}\frac{3\sin 3x}{x}$
Problem 19
$\lim_{x\to0}\frac{\sin^3\frac{x}{2}}{x^3}$
Problem 20
$\lim_{x\to0}\frac{1-\cos x}{x^2}$
Problem 21
$\lim_{x\to1}\frac{(x^2-6x+5)\sin(x-1)}{(x-1)^2}$
Problem 22
$\lim_{x\to0}\frac{\cos x-\cos3x}{x^2}$
Problem 23
$\lim_{x\to0}\frac{\sin^2x}{1-\cos x}$
Problem 24
$\lim_{x\to0}\frac{\sin 5x-\sin 3x}{\sin x}$
Problem 25
$\lim_{x\to2}\frac{\sqrt{7+x}-\sqrt{11-x}}{\sqrt{x+2}-\sqrt{3x-2}}$
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