Chapter 2

Evaluate the limit of the function \(f(x)=\frac{|x-4|}{x-4}, \quad x \neq 4\) \(=0, \quad x=4\) at \(x=4\). Whether the limit exists or not. \\{Ans. \(1,-1\), does not exist

Evaluate the limit of the function \(f(x)=1+x^{2}, \quad 0 \leq x \leq 1\) \(=2-x, \quad x>1\) at \(x=1\). Whether the limit exists or not. \\{Ans. 1,2, does not exist \(\\}\)

\text { If } \begin{aligned} &\begin{aligned} f(x) &=\frac{x-|x|}{x}, \quad x \neq 0 \\ &=2, \quad x=0 \end{aligned}\\\ &\text { show that } \lim _{x \rightarrow 0} f(x) \text { does not exist. } \end{aligned}

If \(\begin{aligned} f(x) &=5 x-4, \quad 0

Let \(\begin{aligned} f(x) &=\cos x, & & x \geq 0 \\ &=x+k, & & x<0 \end{aligned}\) Find the value of constant \(k\), given that \(\lim _{x \rightarrow 0} f(x)\) exists. \\{Ans. \(\left.k=1\right\\}\)

Let \(\begin{aligned} f(x) &=\cos x, & & x \geq 0 \\ &=x+k, & & x<0 \end{aligned}\) Find the value of constant \(k\), given that \(\lim _{x \rightarrow 0} f(x)\) exists. \\{Ans. \(\left.k=1\right\\}\)

\lim _{x \rightarrow-2} \frac{x^{2}+5 x+6}{x+2} \text { \\{Ans. 1\\} }

\left.\lim _{x \rightarrow 5} \frac{2 x^{2}-11 x+5}{4 x^{2}-16 x-20} \text { \\{Ans. } \frac{3}{8}\right\\}

\lim _{x \rightarrow 3} \frac{x^{2}-6 x+9}{2 x^{2}+x-21}\\{\text { Ans. } 0\\}

$$ \lim _{x \rightarrow-3} \frac{x^{3}+27}{2 x^{2}-3 x-27}\left\\{\text { Ans. }-\frac{9}{5}\right\\} $$

$$ \lim _{x \rightarrow \frac{1}{2}} \frac{8 x^{3}-1}{6 x^{2}-5 x+1}\\{\text { Ans. } 6\\} $$

$$ \lim _{x \rightarrow 2} \frac{x^{3}-6 x^{2}+11 x-6}{x^{2}-6 x+8} \text { . \\{Ans. } \frac{1}{2} $$

$$ \lim _{x \rightarrow 2} \frac{x^{3}-5 x^{2}+8 x-4}{x^{3}-3 x^{2}+4} \text { \\{Ans. } \frac{1}{3} $$

$$ \lim _{x \rightarrow 3} \frac{1}{x-3}-\frac{6}{x^{2}-9}\left\\{\text { Ans. } \frac{1}{6}\right. $$

$$ \lim _{x \rightarrow 1} \frac{x^{3}+x^{2}-x-1}{x^{3}-x^{2}-x+1}\\{\text { Ans. }+\infty,-\infty\\} $$

$$ \lim _{x \rightarrow 4} \frac{2 x^{2}-4 x-24}{x^{2}-16}-\frac{1}{4-x}\left\\{\text { Ans. } \frac{13}{8}\right\\} $$

$$ \lim _{x \rightarrow 2} \frac{x^{3}+3 x^{2}-9 x-2}{x^{3}-x-6}\left\\{\text { Ans. } \frac{15}{11}\right\\} $$

$$ \lim _{x \rightarrow 1} \frac{x^{3}-3 x+2}{x^{4}-4 x+3}\left\\{\text { Ans. } \frac{1}{2}\right. $$

$$ \lim _{x \rightarrow 1} \frac{1}{1-x}-\frac{3}{1-x^{3}}\\{\text { Ans. }-1\\} $$

$$ \left.\lim _{x \rightarrow-2} \frac{x^{5}+2 x^{4}+x^{2}+3 x+2}{x^{4}+2 x^{3}+3 x^{2}-5 x-22} \text { \\{Ans. }-\frac{3}{5}\right\\} $$

$$ \lim _{x \rightarrow 2} \frac{1}{x(x-2)^{2}}-\frac{1}{x^{2}-3 x+2}\\{\text { Ans. } \infty\\} $$

$$ \lim _{x \rightarrow-2} \frac{2}{x+2}+\frac{1}{x^{2}-2 x+4}-\frac{24}{x^{3}+8}\left\\{\text { Ans. }-\frac{11}{12}\right\\} $$

$$ \left.\lim _{x \rightarrow 1} \frac{1-\sqrt{x}}{1-x^{2}} \text { \\{Ans. } \frac{1}{4}\right\\} $$

$$ \left.\lim _{x \rightarrow 4} \frac{3-\sqrt{5+x}}{1-\sqrt{5-x}} \text { \\{Ans. }-\frac{1}{3}\right\\} $$

$$ \left.\lim _{x \rightarrow 1} \frac{\sqrt{x+8}-\sqrt{8 x+1}}{\sqrt{5-x}-\sqrt{7 x-3}} \text { \\{ns. } \frac{7}{12}\right\\} $$

$$ \left.\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-\sqrt{4-x}}{4 x} \text { \\{Ans. } \frac{1}{8}\right\\} $$

$$ \lim _{x \rightarrow 7} \frac{2-\sqrt{x-3}}{x^{2}-49}\left\\{\text { Ans. }-\frac{1}{56}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x} \text { \\{Ans. 1\\} } $$

$$ \left.\lim _{x \rightarrow 3} \frac{\sqrt{x^{2}-2 x+6}-\sqrt{x^{2}+2 x-6}}{x^{2}-4 x+3} \text { \\{Ans. }-\frac{1}{3}\right\\} $$

$$ \lim _{x \rightarrow-1} \frac{x+1}{2-\sqrt{4+x+x^{2}}}\\{\text { Ans. } 4\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+1}-1}{\sqrt{x^{2}+16}-4}\\{\text { Ans. } 4\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+1}-1}{\sqrt{x^{2}+9}-3} .\\{\text { Ans. } 3\\} $$

$$ \lim _{x \rightarrow 2} \frac{\sqrt[3]{10-x}-2}{x-2}\left\\{\text { Ans. }-\frac{1}{12}\right\\} $$

$$ \lim _{x \rightarrow 1} \frac{x-1}{\sqrt[3]{26+x}-3}\\{\text { Ans. } 27\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sqrt[3]{1+x^{2}}-1}{x^{2}}\left\\{\text { Ans. } \frac{1}{3}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sqrt[3]{1+x}-\sqrt[3]{1-x}}{x}\left\\{\text { Ans. } \frac{2}{3}\right\\} $$

$$ \lim _{x \rightarrow \infty} \frac{2 x^{2}-4 x+1}{4 x^{2}+6 x-5}\left\\{\text { Ans. } \frac{1}{2}\right. $$

$$ \lim _{x \rightarrow \infty} \frac{(x-1)^{3}}{x^{3}+1}\\{\text { Ans. } 1\\} $$

$$ \lim _{x \rightarrow \infty} \frac{x^{3}}{3 x^{2}-4}-\frac{x^{2}}{3 x+2}\left\\{\text { Ans. } \frac{2}{9}\right. $$

$$ \lim _{x \rightarrow \infty} \frac{x^{3}+x}{x^{4}-3 x^{2}+1}\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow \infty} \frac{3 x^{2}+2}{x^{4}+1}\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow \infty} \frac{x^{3}}{x^{2}+1}-x\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow \infty} \frac{x^{4}-5 x}{3 x-x^{2}+1}\\{\text { Ans. }-\infty\\} $$

$$ \lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+2}}{x-2}\\{\text { Ans. } 1\\} $$

$$ \lim _{x \rightarrow \infty} \frac{\sqrt{1+x^{2}}-\sqrt{4+x^{2}}}{x}\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow \infty} \frac{\sqrt{1+x^{4}}-1-x^{2}}{x^{2}} .\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow \infty} x^{\frac{3}{2}}\left(\sqrt{x^{3}+1}-\sqrt{x^{3}-1}\right) \text { \\{Ans. 1\\} } $$

$$ \lim _{x \rightarrow \infty} \sqrt{x^{2}+1}-\sqrt{x^{2}-1}\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow \infty} \frac{\sqrt{1+9 x^{2}}+\sqrt{x^{2}-1}}{\sqrt{1+9 x^{2}-\sqrt{x^{2}-1}}} \text { \\{Ans. } \frac{4}{3} \text { \\} } $$

$$ \lim _{x \rightarrow \infty} \frac{e^{x}-1}{e^{x}+1}\\{\text { Ans. } 1\\} $$