Problem Set - Limits

## Problem Set - Limits

Last updates: 7 December 2009

## Limits

 Define the following and give an example for each: (a)   continuous at $p$, (b)   $\underset{x\to a}{\mathrm{lim}}f\left(x\right)$, (c)   continuous, (d)   uniformly continuous, (d)   Lipschiz continuous, (e)   derivative at $p$, For each of the following, guess the limit and then prove the guess by using the definition of limit: (a)   $\underset{x\to 4}{\mathrm{lim}}\left(\frac{1}{2}x-3\right)$, (b)   $\underset{x\to 0}{\mathrm{lim}}\frac{1}{1+x}$, (c)   $\underset{x\to 4}{\mathrm{lim}}\frac{1}{1+{x}^{2}}$, (d)   $\underset{x\to 1}{\mathrm{lim}}\frac{{x}^{2}-1}{x-1}$, (e)   $\underset{x\to 9}{\mathrm{lim}}\frac{x+1}{{x}^{2}+1}$, (f)   $\underset{x\to \infty }{\mathrm{lim}}\frac{\mathrm{sin}x}{x}$, (g)   $\underset{x\to 2}{\mathrm{lim}}\frac{2{x}^{2}+3x-8}{{x}^{3}-2{x}^{2}+x-12}$, (h)   $\underset{x\to \infty }{\mathrm{lim}}\frac{\mathrm{log}x+2x}{3x-5}$, Evaluate the following limits: (a)   $\underset{x\to 0}{\mathrm{lim}}x\mathrm{cos}\frac{1}{{x}^{2}}$, (b)   $\underset{x\to 0}{\mathrm{lim}}\left(\sqrt{5+{x}^{2}}-\sqrt{{x}^{-2}-1}\right)$, (c)   $\underset{x\to 0}{\mathrm{lim}}\frac{\sqrt{1+x}-1}{x}$, (d)   $\underset{x\to \infty }{\mathrm{lim}}\frac{{x}^{4}+x}{{x}^{4}+1}$, (e)   $\underset{x\to \infty }{\mathrm{lim}}\frac{7x-1}{{x}^{2}}$, (f)   $\underset{x\to {0}^{+}}{\mathrm{lim}}\frac{\sqrt{x}}{\sqrt{7+\sqrt{x+5}}}$, (g)   $\underset{x\to 1}{\mathrm{lim}}\frac{|x-1|+1}{x+|x+1|}$, (h)   $\underset{x\to \infty }{\mathrm{lim}}\frac{3{x}^{2}+1}{2x+1}$, Evaluate the following limits: (a)   $\underset{x\to 0}{\mathrm{lim}}\frac{1-\mathrm{cos}x}{x+{x}^{2}}$, (b)   $\underset{x\to \infty }{\mathrm{lim}}\frac{\mathrm{log}x}{x}$, (c)   $\underset{x\to 0+}{\mathrm{lim}}\sqrt{x} \mathrm{log}x$, (d)   $\underset{x\to 0+}{\mathrm{lim}}\frac{\sqrt{x}}{\mathrm{log}x}$, (e)   $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}x}{x}$, (f)   $\underset{x\to 0}{\mathrm{lim}}\left(\frac{1}{\mathrm{arcsin}x}-\frac{1}{\mathrm{sin}x}\right)$.