Limits by LHopitals rule Calculator

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Example

Created on 2024-06-20Asked by Alexander Allen (Solvelet student)

Evaluate the limit

\lim_{x \to 0} \frac{\sin x}{x}

using LHopitals rule.

Solution

To evaluate the limit

\lim_{x \to 0} \frac{\sin x}{x}

using L'Hôpital's rule, we recognize that the limit is of the indeterminate form

\frac{0}{0}

. L'Hôpital's rule states that if

\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{0}{0}

\frac{\pm \infty}{\pm \infty}

, then:

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

Here,

f(x) = \sin x

and

g(x) = x

. Thus, we find the derivatives:

f'(x) = \cos x

g'(x) = 1

Now apply L'Hôpital's rule:

\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = \cos 0 = 1

Therefore, the limit is:

1

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Some of the related questions asked by Aria Lee on Solvelet

1. Find the limit

\lim_{x \to 0} \frac{\sin(x)}{x}

using L'Hôpital's rule 2. Find the limit

\lim_{x \to \infty} \frac{x^2}{e^x}

using L'Hopital's rule.

DefinitionCalculates Limits by L'Hôpital's Rule — uses a number of methods related to L'Hôpital's Rule to zero in on the Limit of Indeterminate Forms like (0/0 or ∞/∞) by taking derivatives. Example: Calculate \( \lim_{{x\to0}} \frac{\sin x}{x} \) There few different ways to go about solving this limit, one of which is using L'Hôpital. Note that ( \frac{0}{0} ) is an indeterminate form. [\lim_{{x \to 0}} \frac{\sin x}{x} = \lim_{{x \to 0}} \frac{\cos x}{1} = \cos 0 = 1][5]

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