Learn L’Hôpital’s method to solve indeterminate form function limits easily. Includes example problems and complete solutions!
Solving function limits with the L’Hôpital’s Method is very important for handling indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
🔍 What is L’Hôpital’s Rule?
L’Hôpital’s Rule is a method used to solve indeterminate form limits such as:
- $\frac{0}{0}$
- $\frac{\infty}{\infty}$
With this rule, we can simplify the limit by differentiating the numerator and denominator separately.
📘 General Formula:
If:
$$ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{0}{0} \text{ or } \frac{\infty}{\infty} $$
Then:
$$ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} $$
⚠️ Condition: The functions $f(x)$ and $g(x)$ must be differentiable around $x = a$, and $g'(x) \neq 0$.
✍️ Example Problems and Solutions Using L’Hôpital
🌟 Example 1:
$$ \lim_{x \to 0} \frac{\sin x}{x} $$
Form: $\frac{0}{0}$
Use L’Hôpital:
$$
f(x) = \sin x, \quad g(x) = x \
f'(x) = \cos x, \quad g'(x) = 1
$$
$$ \lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = \boxed{1} $$
🌟 Example 2:
$$ \lim_{x \to \infty} \frac{3x^2 + 5}{2x^2 - x} $$
Direct substitution gives the form $\frac{\infty}{\infty}$
Differentiate numerator and denominator:
$$ f'(x) = 6x, \quad g'(x) = 4x - 1 $$
$$ \lim_{x \to \infty} \frac{6x}{4x - 1} = \lim_{x \to \infty} \frac{6}{4 - \frac{1}{x}} = \boxed{\frac{6}{4} = \frac{3}{2}} $$
🌟 Example 3:
$$ \lim_{x \to 0} \frac{e^x - 1}{x} $$
Form: $\frac{0}{0}$
Differentiate:
$$
f'(x) = e^x, \quad g'(x) = 1 \
\lim_{x \to 0} \frac{e^x}{1} = \boxed{1}
$$
⚠️ Important Notes
- L’Hôpital’s Rule can only be used if direct substitution really results in $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
- If after one differentiation the result is still indeterminate, continue differentiating again (can be two or three times if necessary).
🧠 Practice Problems
- $$\lim_{x \to 0} \frac{\ln(1 + x)}{x}$$
- $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$
- $\lim_{x \to \infty} \frac{x}{\ln x}$
💡 Use L’Hôpital’s Rule and check your answers!
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