• About
• Statistics
• Number Theory
• Java
• Data Structures
• Cornerstones
• Calculus

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

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

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

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

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

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

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

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

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

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

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

   
   

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

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

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

   
   

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

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

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

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

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

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

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

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

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

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

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

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

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

   
   

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

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

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

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

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

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

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

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

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

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

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

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

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