Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Move the limit into the exponent.

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

Step 1.4

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

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

Step 2

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

$\frac{1 - e^{- 2 \cdot 0}}{\sec (x)}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 2$ by $0$ .

$\frac{1 - e^{0}}{\sec (x)}$

Step 3.1.2

Anything raised to $0$ is $1$ .

$\frac{1 - 1 \cdot 1}{\sec (x)}$

Step 3.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{\sec (x)}$

Step 3.1.4

Subtract $1$ from $1$ .

$\frac{0}{\sec (x)}$

Step 3.2

Separate fractions.

$\frac{0}{1} \cdot \frac{1}{\sec (x)}$

Step 3.3

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

$\frac{0}{1} \cdot \frac{1}{\frac{1}{\cos (x)}}$

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Divide $0$ by $1$ .

$0 \cos (x)$

Step 3.7

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

$0$