Exercises: Limits and Continuity

The graphs of $f$ and $g$ are shown in the following figure. Use the limit laws and evaluate the following limits, if they exist.

1. ${\displaystyle \underset{x\to -3}{lim}[3f(x)-2g(x)]}$
2. ${\displaystyle \underset{x\to -3}{lim}\frac{f(x)}{4g(x)}}$
3. ${\displaystyle \underset{x\to 0}{lim}[f(x)g(x)]}$

The graphs of $f$ and $g$ are shown in the following figure. Use the limit laws and evaluate the following limits, if they exist.

1. ${\displaystyle \underset{x\to 1}{lim}[f(x)-g(x)]}$
2. ${\displaystyle \underset{x\to 1}{lim}[f(x)g(x)]}$
3. ${\displaystyle \underset{x\to 1}{lim}\frac{g(x)}{f(x)}}$

Show that ${\displaystyle \underset{x\to 0}{lim}x\lfloor \frac{1}{x}\rfloor =1}$, where $\lfloor \cdot \rfloor$ denotes the floor (greatest integer) function.

[Hint: Use the Sandwich Theorem.]

Evaluate the following limits:

1. ${\displaystyle \underset{x\to 0}{lim}x\sin \left(\frac{1}{x}\right)}$
2. ${\displaystyle \underset{x\to 0^{+}}{lim}\sqrt{x}\cos \left(\frac{1}{x^2}\right)}$
3. ${\displaystyle \underset{x\to 0}{lim}\sin x\sin \left(\frac{1}{x}\right)}$

Show that ${\displaystyle \underset{x\to 0}{lim}\frac{\tan x}{x}=1}$.

Let

For what value of the constant $m$ is $f$ continuous at $x=3$?

For what values of $x$ is there a discontinuity in the graph of

Is $f(x)=\sqrt{(x-1{)}^2(x-2)}$ continuous on its domain?

Where are the following functions continuous?

1. $h(x)=\cos (x^3+1)$
2. $F(x)=\left|{\displaystyle \frac{x\cos x}{x^2+1}}\right|$
3. $G(x)=\sqrt{x^2+2x-2}$

Find ${\displaystyle \underset{x\to 3}{lim}\frac{x^3-27}{x^3-2x^2-5x+6}}$.

Given , find ${\displaystyle \underset{h\to 0}{lim}\frac{\sqrt{x+h}-\sqrt{x}}{h}}$.

Find ${\displaystyle \underset{x\to 2}{lim}\frac{{\displaystyle \frac{1}{x+1}}-{\displaystyle \frac{1}{3}}}{x-2}}$.

Find

Find:

1. ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}\frac{3+5x}{1-2x}}$
2. ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{3x^3-4x^2+x+3}{-x^3+2x^2}}$
3. ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}\frac{15+4x-5x^3+3x^4}{1+16x^2-x^5}}$
4. ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{x^7+1}{-x^6+x^3+9}}$

Find ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}\frac{3x-4}{5x-\sin x^2}}$.

Find ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}(\frac{2x^3}{x^2-4}-\frac{2x^2}{x-1})}$.

Find ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}\frac{x^2-4\sqrt{x}+3x}{-3x^2+5x^{2/3}+1}}$.

[Hint: Factor out the highest power of $x$.]

Find ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{\sqrt{4x^4+3x^2+1}}{1-2x^2}}$.

Find:

1. ${\displaystyle \underset{t\to +\mathrm{\infty }}{lim}\frac{\sqrt{t^2+6}}{3t-9}}$
2. ${\displaystyle \underset{t\to -\mathrm{\infty }}{lim}\frac{\sqrt{t^2+6}}{3t-9}}$

Find:

1. ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}(\sqrt{x^4+5}-2x^2)}$
2. ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}(\sqrt{x^4+5x^2}-x^2)}$
3. ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}(\sqrt{2x^2-1}-7x)}$

Find the vertical asymptotes (if any) of the following functions:

1. $f(x)={\displaystyle \frac{4-2x}{2x^2+x-1}}$
2. $g(x)={\displaystyle \frac{x^2+4x+3}{x^2+7x+12}}$
3. $h(x)={\displaystyle \frac{x+7}{x^2+1}}$

Find the vertical asymptotes of ${\displaystyle f(x)=\frac{x-2}{\sqrt{4-x^2}}}$.

Find the vertical asymptotes of:

1. $F(x)=\ln (x-2)$
2. $g(x)=\ln \left(\sqrt{x^2-9}\right)$

Determine the end behavior of the graph of ${\displaystyle y=\frac{2x^2-3}{x^2+1}}$.

Determine the end behavior of ${\displaystyle f(x)=\frac{1-x}{|x|+3}}$.

Find the oblique asymptote of ${\displaystyle f(x)=-3x+1+\frac{3x+7}{x^4-2x+2}}$.

Show that if the line $y=mx+b$ is the oblique asymptote of $f(x)$ as $x\to +\mathrm{\infty }$, then

If the line is the oblique asymptote as $x\to -\mathrm{\infty }$, replace $x\to +\mathrm{\infty }$ with $x\to -\mathrm{\infty }$ in both equations.

Use the result of the previous exercise to find the oblique asymptote(s) of