Calculus Examples

$lim_{x \to 0} \frac{\cos (x) \tan (x)}{x}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 1.2

Cancel the common factor of $\cos (x)$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 2

The limit of $\frac{\sin (x)}{x}$ as $x$ approaches $0$ is $1$ .

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Move the limit inside the trig function because sine is continuous.

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

Step 2.1.2.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 2.1.2.3

The exact value of $\sin (0)$ is $0$ .

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

Step 2.1.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{0}$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{\sin (x)}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 2.3.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$lim_{x \to 0} \frac{\cos (x)}{\frac{d}{d x} [x]}$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{\cos (x)}{1}$

Step 2.4

Evaluate the limit.

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Step 2.4.1

Divide $\cos (x)$ by $1$ .

$lim_{x \to 0} \cos (x)$

Step 2.4.2

Move the limit inside the trig function because cosine is continuous.

$\cos (lim_{x \to 0} x)$

Step 2.5

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\cos (0)$

Step 2.6

The exact value of $\cos (0)$ is $1$ .

$1$