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Algebra Baldor
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Find the limit of $\left(\frac{1-\cos\left(x\right)}{x}\right)^2$ as $x$ approaches 0

Used Formulas

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Solution

Formulas

Videos

Limits

· Power Rule for Limits
$\lim_{x\to c}\left(a^b\right)={\left(\lim_{x\to c}\left(a\right)\right)}^b$

Basic Differentiation Rules

· Derivative of a Constant
$\frac{d}{dx}\left(c\right)=0$
· Sum Rule for Differentiation
$\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)$
$\frac{d}{dx}\left(cx\right)=c\frac{d}{dx}\left(x\right)$
· Derivative of the linear function
$\frac{d}{dx}\left(x\right)=1$

Derivatives of trigonometric functions

· Derivative of the cosine function
$\frac{d}{dx}\left(\cos\left(\theta \right)\right)=-\sin\left(\theta \right)$

Function Plot

Plotting: $\left(\frac{1-\cos\left(x\right)}{x}\right)^2$

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Find the limit

$\lim_{x\to1}\left(\frac{xe^x-e}{x^2-1}\right)$

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Find the limit

$\lim_{x\to0}\left(\frac{1-\cos\left(x^2\right)}{x^2}\right)$

Find the limit

$\lim_{x\to0}\left(\frac{\cos\left(5x\right)-1}{x^2}\right)$

Main Topic: Limits by L'Hôpital's rule

In mathematics, and more specifically in calculus, L'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms.

Used Formulas

See formulas (6)

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