Calculus Examples

$lim_{x \to 8} \frac{e^{x} - \cos^{2} (x)}{x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} e^{x} - \cos^{2} (x)}{lim_{x \to 8} x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} e^{x} - lim_{x \to 8} \cos^{2} (x)}{lim_{x \to 8} x}$

Step 3

Move the limit into the exponent.

$\frac{e^{lim_{x \to 8} x} - lim_{x \to 8} \cos^{2} (x)}{lim_{x \to 8} x}$

Step 4

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

$\frac{e^{lim_{x \to 8} x} - {(lim_{x \to 8} \cos (x))}^{2}}{lim_{x \to 8} x}$

Step 5

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

$\frac{e^{lim_{x \to 8} x} - \cos^{2} (lim_{x \to 8} x)}{lim_{x \to 8} x}$

Step 6

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{e^{8} - \cos^{2} (lim_{x \to 8} x)}{lim_{x \to 8} x}$

Step 6.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{e^{8} - \cos^{2} (8)}{lim_{x \to 8} x}$

Step 6.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{e^{8} - \cos^{2} (8)}{8}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Rewrite $e^{8}$ as ${(e^{4})}^{2}$ .

$\frac{{(e^{4})}^{2} - \cos^{2} (8)}{8}$

Step 7.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e^{4}$ and $b = \cos (8)$ .

$\frac{(e^{4} + \cos (8)) (e^{4} - \cos (8))}{8}$

Step 7.1.3

Simplify.

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Step 7.1.3.1

Evaluate $\cos (8)$ .

$\frac{(e^{4} + 0.99026806) (e^{4} - \cos (8))}{8}$

Step 7.1.3.2

Add $e^{4}$ and $0.99026806$ .

$\frac{55.5884181 (e^{4} - \cos (8))}{8}$

Step 7.1.3.3

Evaluate $\cos (8)$ .

$\frac{55.5884181 (e^{4} - 1 \cdot 0.99026806)}{8}$

Step 7.1.3.4

Multiply $- 1$ by $0.99026806$ .

$\frac{55.5884181 (e^{4} - 0.99026806)}{8}$

Step 7.1.3.5

Subtract $0.99026806$ from $e^{4}$ .

$\frac{55.5884181 \cdot 53.60788196}{8}$

Step 7.2

Multiply $55.5884181$ by $53.60788196$ .

$\frac{2979.97735619}{8}$

Step 7.3

Divide $2979.97735619$ by $8$ .

$372.49716952$