Calculus Examples

$lim_{x \to 8} \frac{e^{2 x} - 1}{\sin (3 x)}$

Step 1

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

$\frac{lim_{x \to 8} e^{2 x} - 1}{lim_{x \to 8} \sin (3 x)}$

Step 2

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

$\frac{lim_{x \to 8} e^{2 x} - lim_{x \to 8} 1}{lim_{x \to 8} \sin (3 x)}$

Step 3

Move the limit into the exponent.

$\frac{e^{lim_{x \to 8} 2 x} - lim_{x \to 8} 1}{lim_{x \to 8} \sin (3 x)}$

Step 4

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

$\frac{e^{2 lim_{x \to 8} x} - lim_{x \to 8} 1}{lim_{x \to 8} \sin (3 x)}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{e^{2 lim_{x \to 8} x} - 1 \cdot 1}{lim_{x \to 8} \sin (3 x)}$

Step 6

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

$\frac{e^{2 lim_{x \to 8} x} - 1 \cdot 1}{\sin (lim_{x \to 8} 3 x)}$

Step 7

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{e^{2 lim_{x \to 8} x} - 1 \cdot 1}{\sin (3 lim_{x \to 8} x)}$

Step 8

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

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Step 8.1

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

$\frac{e^{2 \cdot 8} - 1 \cdot 1}{\sin (3 lim_{x \to 8} x)}$

Step 8.2

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

$\frac{e^{2 \cdot 8} - 1 \cdot 1}{\sin (3 \cdot 8)}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{{(e^{8})}^{2} - 1 \cdot 1}{\sin (3 \cdot 8)}$

Step 9.1.2

Rewrite $1$ as $1^{2}$ .

$\frac{{(e^{8})}^{2} - 1^{2}}{\sin (3 \cdot 8)}$

Step 9.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e^{8}$ and $b = 1$ .

$\frac{(e^{8} + 1) (e^{8} - 1)}{\sin (3 \cdot 8)}$

Step 9.1.4

Simplify.

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Step 9.1.4.1

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

$\frac{(e^{8} + 1) ({(e^{4})}^{2} - 1)}{\sin (3 \cdot 8)}$

Step 9.1.4.2

Rewrite $1$ as $1^{2}$ .

$\frac{(e^{8} + 1) ({(e^{4})}^{2} - 1^{2})}{\sin (3 \cdot 8)}$

Step 9.1.4.3

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 $b = 1$ .

$\frac{(e^{8} + 1) ((e^{4} + 1) (e^{4} - 1))}{\sin (3 \cdot 8)}$

Step 9.1.4.4

Simplify.

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Step 9.1.4.4.1

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

$\frac{(e^{8} + 1) ((e^{4} + 1) ({(e^{2})}^{2} - 1))}{\sin (3 \cdot 8)}$

Step 9.1.4.4.2

Rewrite $1$ as $1^{2}$ .

$\frac{(e^{8} + 1) ((e^{4} + 1) ({(e^{2})}^{2} - 1^{2}))}{\sin (3 \cdot 8)}$

Step 9.1.4.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e^{2}$ and $b = 1$ .

$\frac{(e^{8} + 1) ((e^{4} + 1) ((e^{2} + 1) (e^{2} - 1)))}{\sin (3 \cdot 8)}$

Step 9.1.4.4.4

Simplify.

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Step 9.1.4.4.4.1

Rewrite $1$ as $1^{2}$ .

$\frac{(e^{8} + 1) ((e^{4} + 1) ((e^{2} + 1) (e^{2} - 1^{2})))}{\sin (3 \cdot 8)}$

Step 9.1.4.4.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e$ and $b = 1$ .

$\frac{(e^{8} + 1) ((e^{4} + 1) ((e^{2} + 1) (e + 1) (e - 1)))}{\sin (3 \cdot 8)}$

$\frac{(e^{8} + 1) ((e^{4} + 1) (e^{2} + 1) (e + 1) (e - 1))}{\sin (3 \cdot 8)}$

$\frac{(e^{8} + 1) (e^{4} + 1) (e^{2} + 1) (e + 1) (e - 1)}{\sin (3 \cdot 8)}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $3$ by $8$ .

$\frac{(e^{8} + 1) (e^{4} + 1) (e^{2} + 1) (e + 1) (e - 1)}{\sin (24)}$

Step 9.2.2

Evaluate $\sin (24)$ .

$\frac{(e^{8} + 1) (e^{4} + 1) (e^{2} + 1) (e + 1) (e - 1)}{0.40673664}$

Step 9.3

Replace $e$ with an approximation.

$\frac{({2.71828182}^{8} + 1) (e^{4} + 1) (e^{2} + 1) (e + 1) (e - 1)}{0.40673664}$

Step 9.4

Raise $2.71828182$ to the power of $8$ .

$\frac{(2980.95798704 + 1) (e^{4} + 1) (e^{2} + 1) (e + 1) (e - 1)}{0.40673664}$

Step 9.5

Add $2980.95798704$ and $1$ .

$\frac{2981.95798704 (e^{4} + 1) (e^{2} + 1) (e + 1) (e - 1)}{0.40673664}$

Step 9.6

Replace $e$ with an approximation.

$\frac{2981.95798704 ({2.71828182}^{4} + 1) (e^{2} + 1) (e + 1) (e - 1)}{0.40673664}$

Step 9.7

Raise $2.71828182$ to the power of $4$ .

$\frac{2981.95798704 (54.59815003 + 1) (e^{2} + 1) (e + 1) (e - 1)}{0.40673664}$

Step 9.8

Add $54.59815003$ and $1$ .

$\frac{2981.95798704 \cdot 55.59815003 (e^{2} + 1) (e + 1) (e - 1)}{0.40673664}$

Step 9.9

Multiply $2981.95798704$ by $55.59815003$ .

$\frac{165791.34755607 (e^{2} + 1) (e + 1) (e - 1)}{0.40673664}$

Step 9.10

Replace $e$ with an approximation.

$\frac{165791.34755607 ({2.71828182}^{2} + 1) (e + 1) (e - 1)}{0.40673664}$

Step 9.11

Raise $2.71828182$ to the power of $2$ .

$\frac{165791.34755607 (7.38905609 + 1) (e + 1) (e - 1)}{0.40673664}$

Step 9.12

Add $7.38905609$ and $1$ .

$\frac{165791.34755607 \cdot 8.38905609 (e + 1) (e - 1)}{0.40673664}$

Step 9.13

Multiply $165791.34755607$ by $8.38905609$ .

$\frac{1390832.91536521 (e + 1) (e - 1)}{0.40673664}$

Step 9.14

Replace $e$ with an approximation.

$\frac{1390832.91536521 (2.71828182 + 1) (e - 1)}{0.40673664}$

Step 9.15

Add $2.71828182$ and $1$ .

$\frac{1390832.91536521 \cdot 3.71828182 (e - 1)}{0.40673664}$

Step 9.16

Multiply $1390832.91536521$ by $3.71828182$ .

$\frac{5171508.75562519 (e - 1)}{0.40673664}$

Step 9.17

Replace $e$ with an approximation.

$\frac{5171508.75562519 (2.71828182 - 1)}{0.40673664}$

Step 9.18

Subtract $1$ from $2.71828182$ .

$\frac{5171508.75562519 \cdot 1.71828182}{0.40673664}$

Step 9.19

Multiply $5171508.75562519$ by $1.71828182$ .

$\frac{8886109.52050758}{0.40673664}$

Step 9.20

Divide $8886109.52050758$ by $0.40673664$ .

$21847329.64630275$