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Chapter – 13
Applications of Derivatives
L’Hôpital’s Rule is a powerful tool in calculus used to evaluate limits that result in indeterminate forms such as $latex \frac{0}{0}$ or $latex \frac{\infty}{\infty}$. The rule states that if you have a limit of the form: $latex \lim_{x \to c} \frac{f(x)}{g(x)}$
and both $latex f(x)$ and $latex g(x)$ approach 0 or $latex \infty$ as $latex x$ approaches $latex c$, then: $latex \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$ provided that the limit on the right side exists or is infinite.
Conditions for Applying L’Hôpital’s Rule
- The original limit must result in an indeterminate form $latex \frac{0}{0}$ or $latex \frac{\infty}{\infty}$.
- Both $latex f(x)$ and $latex g(x)$ must be differentiable near $latex c$, and $latex g'(x) \neq 0$ near $latex c$.
- The limit of the derivatives $latex \lim_{x \to c} \frac{f'(x)}{g'(x)}$ must exist.
Applications of Derivatives Class 12 Mathematics Notes PDF
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The form 0/0 (L Hospital’s rule)
In calculus, the $latex \frac{0}{0}$ form is one of the most common indeterminate forms encountered when evaluating limits. When a limit results in the expression $latex \frac{0}{0}$, it means that both the numerator and the denominator approach zero as the variable approaches a certain value.
This situation is called indeterminate because the limit could potentially resolve to any value depending on the behavior of the functions involved.
L’Hôpital’s Rule is particularly useful for resolving limits that result in the indeterminate form $latex \frac{0}{0}$. If you encounter a limit of the form $latex \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0}$ where both $latex f(x)$ and $latex g(x)$ approach zero as $latex x$ approaches $latex c$, then L’Hôpital’s Rule allows you to take the derivative of the numerator and the derivative of the denominator and then re-evaluate the limit: $latex \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$.
This new limit may no longer be indeterminate and can often be computed directly. If the resulting limit still yields an indeterminate form, L’Hôpital’s Rule can be applied repeatedly until the limit is resolved or it becomes apparent that another method is needed.
Consider the limit $latex \lim_{x \to 0} \frac{\sin(x)}{x}$. As $latex x$ approaches 0, both $latex \sin(x)$ and $latex x$ approach 0, leading to the indeterminate form $latex \frac{0}{0}$. Applying L’Hôpital’s Rule: $latex \lim_{x \to 0} \frac{\sin(x)}{x} = \lim_{x \to 0} \frac{\cos(x)}{1} = \cos(0) = 1$.
Hence, the limit is 1. This example illustrates how L’Hôpital’s Rule effectively resolves the indeterminate $latex \frac{0}{0}$ form.
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