L'Hospital's Rule - AP Calculus AB

1. Find the limit: $\lim_{x \to \infty} \frac{3x^2}{2x^2 + 5}$ using L'Hospital's Rule.

   $\frac{3}{2}$. After applying L'Hospital's Rule: $\lim_{x \to \infty} \frac{6x}{4x} = \frac{6}{4} = \frac{3}{2}$.

2. Find the limit: $\lim_{x \to \infty} \frac{x}{x^2 + 1}$ using L'Hospital's Rule.

   0. Apply L'Hospital's Rule: $\lim_{x \to \infty} \frac{1}{2x} = 0$.

3. Identify the indeterminate form: $\lim_{x \to \infty} \frac{x^2}{x^2 + x}$.

   $\frac{\infty}{\infty}$. Both numerator and denominator approach infinity.

4. Find the derivative of $\ln(1+x)$ for L'Hospital's Rule.

   $\frac{1}{1+x}$. Chain rule derivative of logarithmic function.

5. Which forms does L'Hospital's Rule apply to?

   $\frac{0}{0}$ and $\frac{\infty}{\infty}$ only. These are the only indeterminate forms where L'Hospital's Rule applies.

6. What must be true of $f(x)$ and $g(x)$ in L'Hospital's Rule?

   $f(x)$ and $g(x)$ must be differentiable near $c$. Functions must be differentiable for L'Hospital's Rule to work.

7. List a requirement for L'Hospital's Rule to be valid.

   Both $f(x)$ and $g(x)$ must approach 0 or $\infty$ as $x \to c$. Essential condition for applying the rule correctly.

8. State L'Hospital's Rule for $\frac{\infty}{\infty}$ form.

   $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$ if $\frac{\infty}{\infty}$ form. Same rule applies for $\frac{\infty}{\infty}$ form.

9. Find the derivative of $\tan x$ for L'Hospital's Rule.

   $\sec^2 x$. Derivative of tangent function.

10. Find the limit: $\lim_{x \to 0} \frac{1 - \cos x}{x}$ using L'Hospital's Rule.

    0. Derivative of $1 - \cos x$ is $\sin x$; $\frac{\sin 0}{1} = 0$.

11. What is L'Hospital's Rule used for?

    Determining limits of indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. Applies when direct substitution gives undefined ratios.

12. Find the limit: $\lim_{x \to 0} \frac{\ln(1+x)}{x}$ using L'Hospital's Rule.

    1. Derivative of $\ln(1+x)$ is $\frac{1}{1+x}$; $\frac{\frac{1}{1+0}}{1} = 1$.

13. Find the derivative of $\cos x$ for L'Hospital's Rule.

    $-\sin x$. Basic trigonometric derivative.

14. What must the original limit be for L'Hospital's Rule to apply?

    An indeterminate form. L'Hospital's Rule requires indeterminate forms to be valid.

15. Find the limit using L'Hospital's Rule: $\lim_{x \to 0} \frac{e^x - 1}{x}$.

    1. Derivative of $e^x - 1$ is $e^x$; derivative of $x$ is $1$; $\frac{e^0}{1} = 1$.

16. Find the derivative of $\sec x$ for L'Hospital's Rule.

    $\sec x \tan x$. Derivative of secant function.

17. Find the limit: $\lim_{x \to \infty} \frac{\ln x}{x}$ using L'Hospital's Rule.

    $0$. Logarithmic function grows slower than linear function.

18. Find the limit: $\lim_{x \to 1} \frac{x^3 - 1}{x - 1}$ using L'Hospital's Rule.

    3. Derivative of $x^3 - 1$ is $3x^2$; derivative of $x - 1$ is $1$; $\frac{3(1)^2}{1} = 3$.

19. What type of limit can L'Hospital's Rule simplify?

    Limits that result in $\frac{0}{0}$ or $\frac{\infty}{\infty}$. Summary of L'Hospital's Rule application scope.

20. Find the derivative of $\ln x$ for L'Hospital's Rule.

    $\frac{1}{x}$. Basic logarithmic derivative.

21. Find the limit: $\lim_{x \to 0} \frac{x^2}{\sin x}$ using L'Hospital's Rule.

    0. Apply L'Hospital's Rule: $\lim_{x \to 0} \frac{2x}{\cos x} = 0$.

22. Find the limit: $\lim_{x \to 0} \frac{\tan x}{x}$ using L'Hospital's Rule.

    1. Derivative of $\tan x$ is $\sec^2 x$; $\frac{\sec^2 0}{1} = 1$.

23. Identify the indeterminate form: $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$.

    $\frac{0}{0}$. Both $1 - \cos 0 = 0$ and $0^2 = 0$ give $\frac{0}{0}$.

24. State L'Hospital's Rule for $\frac{0}{0}$ form.

    $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$ if $\frac{0}{0}$ form. Take derivatives of numerator and denominator separately.

25. Define indeterminate form.

    A form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ where limits are not immediately obvious. Forms where direct evaluation is unclear or undefined.

26. Does L'Hospital's Rule apply to $\lim_{x \to 0} \frac{1}{x}$?

    No, it's not an indeterminate form. The limit approaches $\pm\infty$, not an indeterminate form.

27. Identify the indeterminate form: $\lim_{x \to \infty} \frac{x}{e^x}$.

    $\frac{\infty}{\infty}$. Both $x \to \infty$ and $e^x \to \infty$ give $\frac{\infty}{\infty}$.

28. Find the limit: $\lim_{x \to \infty} \frac{x^2}{e^x}$ using L'Hospital's Rule.

    0. Apply L'Hospital's Rule repeatedly; exponential grows faster than polynomial.

29. What is the derivative of $\sin x$ needed for L'Hospital's Rule?

    $\cos x$. Basic trigonometric derivative.

30. Find the limit: $\lim_{x \to \infty} \frac{\ln(x^2)}{x}$ using L'Hospital's Rule.

    0. Use chain rule: derivative of $\ln(x^2)$ is $\frac{2}{x}$; limit is $0$.

31. Find the limit: $\lim_{x \to \infty} \frac{2x + 3}{x + 1}$ using L'Hospital's Rule.

    2. Apply L'Hospital's Rule: $\lim_{x \to \infty} \frac{2}{1} = 2$.

32. Identify the indeterminate form: $\lim_{x \to 0} \frac{\sin x}{x}$.

    $\frac{0}{0}$. Both $\sin 0 = 0$ and $0$ in denominator give $\frac{0}{0}$.

33. Find the derivative of $e^x$ needed for L'Hospital's Rule.

    $e^x$. Basic exponential derivative.

34. Identify the indeterminate form: $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$.

    $\frac{0}{0}$. Factoring gives $(x+1)(x-1)$ in numerator; creates $\frac{0}{0}$.

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