Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Apply trigonometric identities.

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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Step 1.2.1.2.1

Cancel the common factor.

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

Step 1.2.1.2.2

Rewrite the expression.

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.3

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

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \cos (x)$ and $g (x) = \tan (x)$ .

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

Step 3.3

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$lim_{x \to 0} \frac{\cos (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x]}$

Step 3.4

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

$lim_{x \to 0} \frac{\cos (x) \sec^{2} (x) + \tan (x) (- \sin (x))}{\frac{d}{d x} [x]}$

Step 3.5

Simplify.

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Step 3.5.1

Reorder terms.

$lim_{x \to 0} \frac{\sec^{2} (x) \cos (x) - \sin (x) \tan (x)}{\frac{d}{d x} [x]}$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

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

$lim_{x \to 0} \frac{{(\frac{1}{\cos (x)})}^{2} \cos (x) - \sin (x) \tan (x)}{\frac{d}{d x} [x]}$

Step 3.5.2.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$lim_{x \to 0} \frac{\frac{1^{2}}{\cos^{2} (x)} \cos (x) - \sin (x) \tan (x)}{\frac{d}{d x} [x]}$

Step 3.5.2.3

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

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Step 3.5.2.3.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$lim_{x \to 0} \frac{\frac{1^{2}}{\cos (x) \cos (x)} \cos (x) - \sin (x) \tan (x)}{\frac{d}{d x} [x]}$

Step 3.5.2.3.2

Cancel the common factor.

$lim_{x \to 0} \frac{\frac{1^{2}}{\cos (x) \cos (x)} \cos (x) - \sin (x) \tan (x)}{\frac{d}{d x} [x]}$

Step 3.5.2.3.3

Rewrite the expression.

$lim_{x \to 0} \frac{\frac{1^{2}}{\cos (x)} - \sin (x) \tan (x)}{\frac{d}{d x} [x]}$

Step 3.5.2.4

One to any power is one.

$lim_{x \to 0} \frac{\frac{1}{\cos (x)} - \sin (x) \tan (x)}{\frac{d}{d x} [x]}$

Step 3.5.2.5

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

$lim_{x \to 0} \frac{\frac{1}{\cos (x)} - \sin (x) \frac{\sin (x)}{\cos (x)}}{\frac{d}{d x} [x]}$

Step 3.5.2.6

Multiply $- \sin (x) \frac{\sin (x)}{\cos (x)}$ .

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Step 3.5.2.6.1

Combine $\frac{\sin (x)}{\cos (x)}$ and $\sin (x)$ .

$lim_{x \to 0} \frac{\frac{1}{\cos (x)} - \frac{\sin (x) \sin (x)}{\cos (x)}}{\frac{d}{d x} [x]}$

Step 3.5.2.6.2

Raise $\sin (x)$ to the power of $1$ .

$lim_{x \to 0} \frac{\frac{1}{\cos (x)} - \frac{\sin^{1} (x) \sin (x)}{\cos (x)}}{\frac{d}{d x} [x]}$

Step 3.5.2.6.3

Raise $\sin (x)$ to the power of $1$ .

$lim_{x \to 0} \frac{\frac{1}{\cos (x)} - \frac{\sin^{1} (x) \sin^{1} (x)}{\cos (x)}}{\frac{d}{d x} [x]}$

Step 3.5.2.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 0} \frac{\frac{1}{\cos (x)} - \frac{{\sin (x)}^{1 + 1}}{\cos (x)}}{\frac{d}{d x} [x]}$

Step 3.5.2.6.5

Add $1$ and $1$ .

$lim_{x \to 0} \frac{\frac{1}{\cos (x)} - \frac{\sin^{2} (x)}{\cos (x)}}{\frac{d}{d x} [x]}$

Step 3.5.3

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{1 - \sin^{2} (x)}{\cos (x)}}{\frac{d}{d x} [x]}$

Step 3.5.4

Apply pythagorean identity.

$lim_{x \to 0} \frac{\frac{\cos^{2} (x)}{\cos (x)}}{\frac{d}{d x} [x]}$

Step 3.5.5

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

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Step 3.5.5.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 3.5.5.2

Cancel the common factors.

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Step 3.5.5.2.1

Multiply by $1$ .

$lim_{x \to 0} \frac{\frac{\cos (x) \cos (x)}{\cos (x) \cdot 1}}{\frac{d}{d x} [x]}$

Step 3.5.5.2.2

Cancel the common factor.

$lim_{x \to 0} \frac{\frac{\cos (x) \cos (x)}{\cos (x) \cdot 1}}{\frac{d}{d x} [x]}$

Step 3.5.5.2.3

Rewrite the expression.

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

Step 3.5.5.2.4

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

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

Step 3.6

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 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

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

$\cos (0)$

Step 6

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

$1$