Calculus Examples

$lim_{x \to 0} \frac{2 x}{\tan (x)}$

Step 1

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to 0} \frac{x}{\tan (x)}$

Step 2

Apply L'Hospital's rule.

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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.

$2 \frac{lim_{x \to 0} x}{lim_{x \to 0} \tan (x)}$

Step 2.1.2

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

$2 \frac{0}{lim_{x \to 0} \tan (x)}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

$2 \frac{0}{\tan (lim_{x \to 0} x)}$

Step 2.1.3.2

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

$2 \frac{0}{\tan (0)}$

Step 2.1.3.3

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

$2 (\frac{0}{0})$

Step 2.1.3.4

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

Undefined

$2 (\frac{0}{0})$

Step 2.1.4

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

Undefined

$2 (\frac{0}{0})$

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

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

$2 lim_{x \to 0} \frac{1}{\sec^{2} (x)}$

Step 3

Evaluate the limit.

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Step 3.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$2 \frac{lim_{x \to 0} 1}{lim_{x \to 0} \sec^{2} (x)}$

Step 3.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$2 \frac{1}{lim_{x \to 0} \sec^{2} (x)}$

Step 3.3

Move the exponent $2$ from $\sec^{2} (x)$ outside the limit using the Limits Power Rule.

$2 \frac{1}{{(lim_{x \to 0} \sec (x))}^{2}}$

Step 3.4

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

$2 \frac{1}{\sec^{2} (lim_{x \to 0} x)}$

Step 4

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

$2 \frac{1}{\sec^{2} (0)}$

Step 5

Simplify the answer.

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Step 5.1

Rewrite $1$ as $1^{2}$ .

$2 \frac{1^{2}}{\sec^{2} (0)}$

Step 5.2

Rewrite $\frac{1^{2}}{\sec^{2} (0)}$ as ${(\frac{1}{\sec (0)})}^{2}$ .

$2 {(\frac{1}{\sec (0)})}^{2}$

Step 5.3

Rewrite $\sec (0)$ in terms of sines and cosines.

$2 {(\frac{1}{\frac{1}{\cos (0)}})}^{2}$

Step 5.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (0)}$ .

$2 {(1 \cos (0))}^{2}$

Step 5.5

Multiply $\cos (0)$ by $1$ .

$2 \cos^{2} (0)$

Step 5.6

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

$2 \cdot 1^{2}$

Step 5.7

One to any power is one.

$2 \cdot 1$

Step 5.8

Multiply $2$ by $1$ .

$2$