Calculus Examples

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

Step 1

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

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

Step 2

Move the limit into the exponent.

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

Step 3

Move the limit into the exponent.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Anything raised to $0$ is $1$ .

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

Step 8.1.2

Anything raised to $0$ is $1$ .

$\frac{1 + 1 - 0^{2} - 1 \cdot 2}{\sin^{2} (x) - x^{2}}$

Step 8.1.3

Raising $0$ to any positive power yields $0$ .

$\frac{1 + 1 - 0 - 1 \cdot 2}{\sin^{2} (x) - x^{2}}$

Step 8.1.4

Multiply $- 1$ by $0$ .

$\frac{1 + 1 + 0 - 1 \cdot 2}{\sin^{2} (x) - x^{2}}$

Step 8.1.5

Multiply $- 1$ by $2$ .

$\frac{1 + 1 + 0 - 2}{\sin^{2} (x) - x^{2}}$

Step 8.1.6

Add $1$ and $1$ .

$\frac{2 + 0 - 2}{\sin^{2} (x) - x^{2}}$

Step 8.1.7

Add $2$ and $0$ .

$\frac{2 - 2}{\sin^{2} (x) - x^{2}}$

Step 8.1.8

Subtract $2$ from $2$ .

$\frac{0}{\sin^{2} (x) - x^{2}}$

Step 8.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (x)$ and $b = x$ .

$\frac{0}{(\sin (x) + x) (\sin (x) - x)}$

Step 8.3

Divide $0$ by $(\sin (x) + x) (\sin (x) - x)$ .

$0$