L'Hôpital's Rule and Indeterminate Forms
When calculating some limits, it may lead to indeterminate forms which means evaluating the limit directly is not possible. Some of these forms can be worked around by rearranging, and for some we can use the rule of l'hopital.
Indeterminate Forms
The two main indeterminate forms that we can use l'hopital on are:
- $\frac{0}{0}$: This occurs when both the numerator and denominator approach zero as $x$ approaches a certain value.
- $\frac{\infty}{\infty}$: This happens when both the numerator and denominator grow without bound as $x$ approaches a certain value or infinity.
Other indeterminate forms exist, such as $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^\infty$, and $\infty^0$, but not all can be used with l'hopital
But some forms can be rearranged to match the ones listed above!
L'Hôpital's Rule
L'Hôpital's Rule states that for functions $f(x)$ and $g(x)$ that are differentiable near a point $a$, if the limit $\lim_{x \to a} \frac{f(x)}{g(x)}$ results in an indeterminate form of type $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then:
$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$
provided that the limit of the ratio of derivatives exists or is infinite.
TipL'Hôpital's Rule can also be applied when $a$ is $\infty$ or $-\infty$.
Applying L'Hôpital's Rule
To apply L'Hôpital's Rule:
- Verify that the limit results in an indeterminate form ($\frac{0}{0}$ or $\frac{\infty}{\infty}$).
- Differentiate both the numerator and denominator separately.
- Take the limit of the ratio of these derivatives.
- If the result is still indeterminate, repeat steps 2 and 3 until a determinate form is reached.
Let's evaluate $\lim_{x \to 0} \frac{\sin x}{x}$:
- As $x \to 0$, both $\sin x \to 0$ and $x \to 0$, giving the indeterminate form $\frac{0}{0}$.
- Apply L'Hôpital's Rule: $$\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{(\sin x)'}{(x)'} = \lim_{x \to 0} \frac{\cos x}{1} = 1$$
Thus, $\lim_{x \to 0} \frac{\sin x}{x} = 1$.
