Problems based on L'Hospital's Rule

L'Hospital's Rule is a powerful tool in calculus for evaluating limits that present an indeterminate form, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$. This rule simplifies the process of finding limits that would otherwise be difficult or impossible to evaluate using standard algebraic techniques.

Understanding L'Hospital's Rule

L'Hospital's Rule states that if the functions $f(x)$ and $g(x)$ are differentiable and $g'(x) \neq 0$ near a point $c$ (except possibly at $c$), and if $$ \lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0 \quad \text{or} \quad \pm\infty, $$ then the limit of the ratio of the functions as $x$ approaches $c$ can be found by taking the limit of the ratio of their derivatives: $$ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}, $$ provided that the limit on the right side exists or is $\pm\infty$.

Important Points to Remember

• L'Hospital's Rule can only be applied when the limit presents an indeterminate form.
• The derivatives $f'(x)$ and $g'(x)$ must exist near the point $c$.
• The rule can be applied repeatedly if the limit remains indeterminate after the first application.
• L'Hospital's Rule can be applied to one-sided limits as well.

Differences and Important Points

Aspect	Description
Applicability	Only for indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
Differentiability	Functions must be differentiable near the point of interest.
Non-zero Denominator	$g'(x)$ must not be zero near the point $c$.
Existence of Limit	The limit of $\frac{f'(x)}{g'(x)}$ must exist or be $\pm\infty$.
Repeated Application	Can be used multiple times if necessary.
One-sided Limits	Applicable to both left-hand and right-hand limits.

Examples

Example 1: Basic Application

Evaluate the limit: $$ \lim_{x \to 0} \frac{\sin(x)}{x}. $$

Solution:

Both the numerator and the denominator approach 0 as $x$ approaches 0, which is an indeterminate form $\frac{0}{0}$. Applying L'Hospital's Rule: $$ \lim_{x \to 0} \frac{\sin(x)}{x} = \lim_{x \to 0} \frac{\cos(x)}{1} = \cos(0) = 1. $$

Example 2: Repeated Application

Evaluate the limit: $$ \lim_{x \to 0} \frac{e^x - 1 - x}{x^2}. $$

Solution:

This is an indeterminate form $\frac{0}{0}$. Applying L'Hospital's Rule once: $$ \lim_{x \to 0} \frac{e^x - 1 - x}{x^2} = \lim_{x \to 0} \frac{e^x - 1}{2x}. $$

Still indeterminate, apply L'Hospital's Rule again: $$ \lim_{x \to 0} \frac{e^x - 1}{2x} = \lim_{x \to 0} \frac{e^x}{2} = \frac{1}{2}. $$

Example 3: Infinite Limits

Evaluate the limit: $$ \lim_{x \to \infty} \frac{x}{e^x}. $$

Solution:

This is an indeterminate form $\frac{\infty}{\infty}$. Applying L'Hospital's Rule: $$ \lim_{x \to \infty} \frac{x}{e^x} = \lim_{x \to \infty} \frac{1}{e^x} = 0. $$

Example 4: One-sided Limits

Evaluate the limit: $$ \lim_{x \to 0^+} \frac{\ln(x)}{x}. $$

Solution:

This is an indeterminate form $\frac{-\infty}{0}$. Applying L'Hospital's Rule: $$ \lim_{x \to 0^+} \frac{\ln(x)}{x} = \lim_{x \to 0^+} \frac{1/x}{1} = \lim_{x \to 0^+} \frac{1}{x} = -\infty. $$

Conclusion

L'Hospital's Rule is a valuable technique for evaluating limits involving indeterminate forms. It simplifies complex limit problems by allowing us to focus on the behavior of derivatives rather than the original functions. However, it is crucial to ensure that the rule's conditions are met before applying it, and to remember that it is not a universal solution for all limit problems.