Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Move the exponent $2$ from $\sec^{2} (e^{x} - 1)$ outside the limit using the Limits Power Rule.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Move the limit into the exponent.

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

Step 1.5

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Anything raised to $0$ is $1$ .

$\sec^{2} (1 - 1 \cdot 1)$

Step 3.1.2

Multiply $- 1$ by $1$ .

$\sec^{2} (1 - 1)$

Step 3.2

Subtract $1$ from $1$ .

$\sec^{2} (0)$

Step 3.3

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

$1^{2}$

Step 3.4

One to any power is one.

$1$