# Exercises - Infinite Limits and Limits at Infinity

1. Find all vertical asymptotes associated with the graph of each function

1. $\displaystyle{f\,(x) = \frac{x^2-9}{x^2-4}}$

2. $\displaystyle{f\,(x) = \frac{1}{x^2-2x}}$

3. $\displaystyle{f\,(x) = \frac{x^2+7x+10}{x^2-x-6}}$

4. $\displaystyle{f\,(x) = \frac{x}{(x+4)^2}}$

5. $\displaystyle{f\,(x) = \frac{x^2}{(x+4)^2}}$

2. Find all horizontal asymptotes associated with the graph of each function

1. $\displaystyle{f\,(x) = \frac{x^2-9}{x^2-4}}$

2. $\displaystyle{f\,(x) = \frac{1}{x^2-2x}}$

3. $\displaystyle{f\,(x) = \frac{3x^3-1}{x^3}}$

4. $\displaystyle{f\,(x) = \frac{x^3-4}{x^2+8}}$

3. Evaluate the following limits

1. $\displaystyle{\lim_{x \rightarrow 0^+} \frac{1}{x}}$

2. $\displaystyle{\lim_{x \rightarrow 0^-} \frac{1}{x}}$

3. $\displaystyle{\lim_{x \rightarrow \infty} \frac{1}{x}}$

4. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{1}{x}}$

5. $\displaystyle{\lim_{x \rightarrow \infty} \frac{x^3}{1-x^4}}$

6. $\displaystyle{\lim_{x \rightarrow \infty} \frac{x^3}{1-x^3}}$

7. $\displaystyle{\lim_{x \rightarrow \infty} \frac{x^3}{1-x^2}}$

8. $\displaystyle{\lim_{x \rightarrow \pi/2^+} \tan x}$

9. $\displaystyle{\lim_{x \rightarrow 0^-} \csc x}$

10. $\displaystyle{\lim_{x \rightarrow -\infty} \textrm{Arctan } 2x}$

11. $\displaystyle{\lim_{x \rightarrow \infty} 2 \textrm{ Arctan } x}$

12. $\displaystyle{\lim_{x \rightarrow \infty} \textrm{ Arcsin} \frac{1}{x}}$

1. $\displaystyle{\lim_{x \rightarrow \infty} \ln \left( \frac{1}{x} \right)}$

2. $\displaystyle{\lim_{x \rightarrow \infty} \frac{x^3}{1-x^2}}$

3. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{x^2}{4-x^2}}$

4. $\displaystyle{\lim_{x \rightarrow \infty} \frac{x}{x^2-4}}$

5. $\displaystyle{\lim_{x \rightarrow 1^-} \frac{x^3}{1-x^3}}$

6. $\displaystyle{\lim_{x \rightarrow -2} \frac{x^2}{x^2+4}}$

7. $\displaystyle{\lim_{x \rightarrow 3^-} \frac{|x^2-9|}{x-3}}$

8. $\displaystyle{\lim_{x \rightarrow 0^+} \csc x}$

9. $\displaystyle{\lim_{x \rightarrow \pi/4^-} \cot (2x)}$

10. $\displaystyle{\lim_{x \rightarrow \pi} \cos (\frac{1}{3} x)}$

4. Find the following limits, and then sketch the graph of $y=e^{1/x}$

1. $\displaystyle{\lim_{x \rightarrow \infty} e^{1/x}}$

2. $\displaystyle{\lim_{x \rightarrow -\infty} e^{1/x}}$

1. $\displaystyle{\lim_{x \rightarrow 0^-} e^{1/x}}$

2. $\displaystyle{\lim_{x \rightarrow 0^+} e^{1/x}}$

5. Evaluate the following limits

1. $\displaystyle{\lim_{x \rightarrow \infty} \frac{x^2+7x+10}{x^2-x-6}}$

2. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{x^2-2x+1}{x+1}}$

3. $\displaystyle{\lim_{x \rightarrow \infty} \frac{2x^2-3}{x^2+1}}$

4. $\displaystyle{\lim_{x \rightarrow -3^+} \frac{x^3}{x^2-9}}$

5. $\displaystyle{\lim_{x \rightarrow 3} \frac{\sqrt{x^2-5} - 2}{x-3}}$

6. $\displaystyle{\lim_{x \rightarrow -2^+} \frac{x^3-3x}{x+2}}$

7. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{-3x^3+x^2}{x^3-5}}$

8. $\displaystyle{\lim_{x \rightarrow 4^-} \frac{x^3-3x^2-4x}{x^3-4x^2+x-4}}$

9. $\displaystyle{\lim_{x \rightarrow -2^+} \frac{x^3-3x}{x+2}}$

10. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{3-2x^3}{x^2+1}}$

11. $\displaystyle{\lim_{x \rightarrow 2\pi^+} \csc x}$

12. $\displaystyle{\lim_{x \rightarrow -\infty} 2 \textrm{ Arctan } \frac{x}{2}}$

1. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{-3x^3+x^2}{x^3-5}}$

2. $\displaystyle{\lim_{x \rightarrow \infty} \textrm{ Arctan } \frac{1}{x}}$

3. $\displaystyle{\lim_{x \rightarrow \infty} \ln \frac{1}{x}}$

4. $\displaystyle{\lim_{x \rightarrow \infty} e^{1/x}}$

5. $\displaystyle{\lim_{x \rightarrow 1^-} \frac{e^x}{x-1}}$

6. $\displaystyle{\lim_{x \rightarrow 0^-} \ln (x^2)}$

7. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{4-x^3}{x^2+2x}}$

8. $\displaystyle{\lim_{x \rightarrow \infty} \frac{2x^2-4}{5-x^2}}$

9. $\displaystyle{\lim_{x \rightarrow -1} \textrm{Arcsin } \frac{x}{2}}$

10. $\displaystyle{\lim_{x \rightarrow 2^+} \frac{x^2-4}{x-2}}$

11. $\displaystyle{\lim_{x \rightarrow 2^-} \frac{x^3+8}{x^2-4}}$

12. $\displaystyle{\lim_{x \rightarrow -\infty} 2 \textrm{ Arctan } \frac{x}{2}}$

6. Evaluate the following limits

1. $\displaystyle{\lim_{x \rightarrow \infty} \frac{4-x^3}{2x^2+1}}$

2. $\displaystyle{\lim_{x \rightarrow -\infty} e^{1/2x}}$

3. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{4-3x^2}{x^2-1}}$

4. $\displaystyle{\lim_{x \rightarrow 1^-} \frac{4-3x^2}{x^2-1}}$

5. $\displaystyle{\lim_{x \rightarrow 4} \frac{x^3-64}{x^2-16}}$

6. $\displaystyle{\lim_{x \rightarrow -\infty} \textrm{Arcsin } \left(\frac{1}{x} - \frac{1}{2} \right)}$

7. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{1}{e^x}}$

8. $\displaystyle{\lim_{x \rightarrow \infty} \frac{3-e^x}{2e^x-1}}$

9. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{x^2-x}{3-2x^2}}$

10. $\displaystyle{\lim_{x \rightarrow -2^+} \frac{x^3-8}{x^2-4}}$

11. $\displaystyle{\lim_{x \rightarrow 0^+} \frac{1}{x^2-x}}$

12. $\displaystyle{\lim_{x \rightarrow \infty} \frac{x^3}{x^2-4}}$

13. $\displaystyle{\lim_{x \rightarrow \pi^-} \csc x}$

1. $\displaystyle{\lim_{x \rightarrow 1^-} \frac{x^2}{x+1}}$

2. $\displaystyle{\lim_{x \rightarrow \infty} \frac{1-x^2}{3x^2}}$

3. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{1-x^3}{3x^2}}$

4. $\displaystyle{\lim_{x \rightarrow 0} \tan x}$

5. $\displaystyle{\lim_{x \rightarrow -3^+} \frac{|3-x|}{x^2-9}}$

6. $\displaystyle{\lim_{x \rightarrow 2^-} \frac{x^3}{2x^2-8}}$

7. $\displaystyle{\lim_{x \rightarrow -2^-} \frac{x^3}{2x^2-8}}$

8. $\displaystyle{\lim_{x \rightarrow -\infty} \frac{1-x^3}{x^2+2}}$

9. $\displaystyle{\lim_{x \rightarrow \infty} \frac{1}{x}}$

10. $\displaystyle{\lim_{x \rightarrow \infty} \frac{2-3x^2}{x^2-2x}}$

11. $\displaystyle{\lim_{x \rightarrow 2^-} \frac{x}{x^2-4}}$

12. $\displaystyle{\lim_{x \rightarrow -3^+} \frac{x^3}{x^2-9}}$