L'Hopital's Rule Supplemental Exercises

Directions: Find the limit. Be careful! L'Hopital's rule is not always needed and may not (directly) apply!

Note: Clicking on ALPHA will take you to Wolfram Alpha's solution. Clicking on SYMBO will take you to Symbolab's solution. While Symbolab will show step-by-step solutions, it cannot solve all of these problems (at least at the time of the writing of this handout).

  1. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 3} \dfrac{x-3}{3x^2-13x+12}\qquad \qquad$

  2. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 0} \dfrac{2\cosh(x)-2}{1-\cos(2x)}\qquad \qquad$

  3. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 0} \dfrac{x-\mathrm{arctan}(x)}{x^3}\qquad \qquad$

  4. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 0} \dfrac{x+\tan(x)}{\sin(x)}\qquad \qquad$

  5. ALPHA     SYMBO $\qquad \qquad \lim\limits_{t \to 0} \dfrac{5^t-3^t}{t}\qquad \qquad$

  6. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 1} \dfrac{\ln(x)}{x-1}\qquad \qquad$

  7. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to \infty} \dfrac{\ln(x)}{\sqrt[3]{x}}\qquad \qquad$

  8. ALPHA     SYMBO $\qquad \qquad \lim\limits_{t \to 0} \dfrac{\sin(6x)}{\sinh(2x)}\qquad \qquad$

  9. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 0} \dfrac{e^x-1-x-x^2/2}{x^3}\qquad \qquad$

  10. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 0} \dfrac{\mathrm{arcsin}(x)}{x}\qquad \qquad$

  11. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 0} \dfrac{1-e^{2x}}{\sec(x)}\qquad \qquad$

  12. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 2} \dfrac{\ln(5x-9)}{x^3-8}\qquad \qquad$

  13. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 1} \dfrac{\ln(x)}{\tan(\pi x)}\qquad \qquad$

  14. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to -\infty} x^2e^x\qquad \qquad$

  15. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 0^+} \sin(x) \ln(x)\qquad \qquad$

  16. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to \infty} x-\ln(x)\qquad \qquad$

  17. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to \infty} x^{1/x}\qquad \qquad$

  18. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 0^+} x^{1/x}\qquad \qquad$

  19. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 0^+} (\cos(x))^{1/x^2}\qquad \qquad$

  20. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 0} (1+x)^{1/x}\qquad \qquad$

  21. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to 0^+} x^{\sin(x)}\qquad \qquad$

  22. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to (\pi/2)^-} (\tan(x))^{\cos(x)}\qquad \qquad$

  23. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to \infty} (\ln(x))^{1/x}\qquad \qquad$

  24. ALPHA     SYMBO $\qquad \qquad \lim\limits_{x \to \infty} (3^x+5^x)^{1/x}\qquad \qquad$

Answers:
  1. $\lim\limits_{x \to 3} \dfrac{x-3}{3x^2-13x+12}=1/5$

  2. $\lim\limits_{x \to 0} \dfrac{2\cosh(x)-2}{1-\cos(2x)}=1/2$

  3. $\lim\limits_{x \to 0} \dfrac{x-\mathrm{arctan}(x)}{x^3}=1/3$

  4. $\lim\limits_{x \to 0} \dfrac{x+\tan(x)}{\sin(x)}=2$

  5. $\lim\limits_{t \to 0} \dfrac{5^t-3^t}{t}=\ln(5/3)$

  6. $\lim\limits_{x \to 1} \dfrac{\ln(x)}{x-1}=1$

  7. $\lim\limits_{x \to \infty} \dfrac{\ln(x)}{\sqrt[3]{x}}=0$

  8. $\lim\limits_{t \to 0} \dfrac{\sin(6x)}{\sinh(2x)}=3$

  9. $\lim\limits_{x \to 0} \dfrac{e^x-1-x-x^2/2}{x^3}=1/6$

  10. $\lim\limits_{x \to 0} \dfrac{\mathrm{arcsin}(x)}{x}=1$

  11. $\lim\limits_{x \to 0} \dfrac{1-e^{2x}}{\sec(x)}=0$

  12. $\lim\limits_{x \to 2} \dfrac{\ln(5x-9)}{x^3-8}=5/12$

  13. $\lim\limits_{x \to 1} \dfrac{\ln(x)}{\tan(\pi x)}=1/\pi$

  14. $\lim\limits_{x \to -\infty} x^2e^x=0$

  15. $\lim\limits_{x \to 0^+} \sin(x) \ln(x)=0$

  16. $\lim\limits_{x \to \infty} x-\ln(x)=\infty$

  17. $\lim\limits_{x \to \infty} x^{1/x}=1$

  18. $\lim\limits_{x \to 0^+} x^{1/x}=0$

  19. $\lim\limits_{x \to 0^+} (\cos(x))^{1/x^2}=1/\sqrt{e}$

  20. $\lim\limits_{x \to 0} (1+x)^{1/x}=e$

  21. $\lim\limits_{x \to 0^+} x^{\sin(x)}=1$

  22. $\lim\limits_{x \to (\pi/2)^-} (\tan(x))^{\cos(x)}=1$

  23. $\lim\limits_{x \to \infty} (\ln(x))^{1/x}=1$

  24. $\lim\limits_{x \to \infty} (3^x+5^x)^{1/x}=5$