# Limits of Functions: Problems with Solutions

Problem 1
Select the value of the limit $$\lim_{x\rightarrow 0} \frac{1}{x}\times \left(\frac{1}{x+4}-\frac{1}{4}\right)$$
Problem 2
Select the value of the limit $$\lim_{x\rightarrow 4} \frac{1}{x-4}\times\left(\frac{1}{x}-\frac{1}{4}\right)$$
Problem 3
Select the value of the limit $$\lim_{x\rightarrow 2} \left(1-\frac{2}{x}\right)\times \left(\frac{3}{4-x^2}\right)$$
Problem 4
Find the limit: $$\lim_{x\rightarrow -2} \frac{(x^2-x-6)^2}{(x+2)^2}$$
Problem 5
Select the value of the limit $$\lim_{x\rightarrow 5} \frac{\sqrt{x^2+5}-\sqrt{30}}{x-5}$$.
Problem 6
Find the value of $$\lim_{x\rightarrow -3^-} \frac{x+3}{\sqrt{x^2-12x-45}}$$
Problem 7
Find the value of $$\lim_{x\rightarrow 8} \frac{\sqrt[4]{x}-\sqrt[4]{8}}{\sqrt{x}-\sqrt{8}}$$
Problem 8
Find the value of $$\lim_{h\rightarrow 0} \frac{1-\frac{1}{h^2}}{1-\frac{1}{h}}$$.
Problem 9
Find the value of $$\lim_{x\rightarrow 1} \left( \lfloor\frac{x-1}{2}\rfloor + \lfloor\frac{x+1}{2}\rfloor \right)$$
Problem 10
Select the limit of the function: $$\lim_{x\rightarrow 2} \frac{\sqrt{x^2+5}-3}{x^2-2x}$$.

Problem 11
Select the value of the limit $$\lim_{h\rightarrow 1} \frac{\sqrt{b+2(h-1)}-\sqrt{b}}{h-1}$$.
b is a constant.
Problem 12
Find the value of $$\lim_{x\rightarrow 0} \frac{x \sin x}{\sqrt{x^2+1}-1}$$
Problem 13
Find the limit $$\lim_{x\rightarrow 1^{-}} \frac{(\cos^{-1} x)^2}{1-x^2}$$
Problem 14
Select the value of the limit $$\lim_{x\rightarrow 2} \frac{(x^2-x-2)^{40}}{(x^3-12x+16)^{20}}$$
Problem 15
Find the value of the limit $$\lim_{x\rightarrow 2^{-}} \frac{(x+1)^3-27}{|x-2|}$$

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