Calculus Examples

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

Step 1

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

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

Step 2

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

Move the limit into the exponent.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

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

Step 8

Move the limit into the exponent.

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

Step 9

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

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

Step 10

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $2$ by $0$ .

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

Step 11.1.2

Anything raised to $0$ is $1$ .

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

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $2$ by $0$ .

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

Step 11.2.2

Anything raised to $0$ is $1$ .

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

Step 11.2.3

Multiply $- 1$ by $1$ .

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

Step 11.2.4

Subtract $1$ from $2$ .

$\frac{- 1 \cdot 1}{1^{2}}$

Step 11.2.5

One to any power is one.

$\frac{- 1 \cdot 1}{1}$

Step 11.3

Multiply $- 1$ by $1$ .

$\frac{- 1}{1}$

Step 11.4

Divide $- 1$ by $1$ .

$- 1$