Calculus Examples

$lim_{x \to 8} \frac{3 e^{\frac{5}{x}} - 3}{π - 2 \arctan (x)}$

Step 1

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

$\frac{lim_{x \to 8} 3 e^{\frac{5}{x}} - 3}{lim_{x \to 8} π - 2 \arctan (x)}$

Step 2

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

$\frac{lim_{x \to 8} 3 e^{\frac{5}{x}} - lim_{x \to 8} 3}{lim_{x \to 8} π - 2 \arctan (x)}$

Step 3

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

$\frac{3 lim_{x \to 8} e^{\frac{5}{x}} - lim_{x \to 8} 3}{lim_{x \to 8} π - 2 \arctan (x)}$

Step 4

Move the limit into the exponent.

$\frac{3 e^{lim_{x \to 8} \frac{5}{x}} - lim_{x \to 8} 3}{lim_{x \to 8} π - 2 \arctan (x)}$

Step 5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 e^{5 lim_{x \to 8} \frac{1}{x}} - lim_{x \to 8} 3}{lim_{x \to 8} π - 2 \arctan (x)}$

Step 6

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

$\frac{3 e^{5 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x}} - lim_{x \to 8} 3}{lim_{x \to 8} π - 2 \arctan (x)}$

Step 7

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

$\frac{3 e^{5 \frac{1}{lim_{x \to 8} x}} - lim_{x \to 8} 3}{lim_{x \to 8} π - 2 \arctan (x)}$

Step 8

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$\frac{3 e^{5 \frac{1}{lim_{x \to 8} x}} - 1 \cdot 3}{lim_{x \to 8} π - 2 \arctan (x)}$

Step 9

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

$\frac{3 e^{5 \frac{1}{lim_{x \to 8} x}} - 1 \cdot 3}{lim_{x \to 8} π - lim_{x \to 8} 2 \arctan (x)}$

Step 10

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

$\frac{3 e^{5 \frac{1}{lim_{x \to 8} x}} - 1 \cdot 3}{π - lim_{x \to 8} 2 \arctan (x)}$

Step 11

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

$\frac{3 e^{5 \frac{1}{lim_{x \to 8} x}} - 1 \cdot 3}{π - 2 lim_{x \to 8} \arctan (x)}$

Step 12

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

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Step 12.1

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

$\frac{3 e^{5 (\frac{1}{8})} - 1 \cdot 3}{π - 2 lim_{x \to 8} \arctan (x)}$

Step 12.2

Evaluate the limit of $\arctan (x)$ by plugging in $8$ for $x$ .

$\frac{3 e^{5 (\frac{1}{8})} - 1 \cdot 3}{π - 2 \arctan (8)}$

Step 12.3

Evaluate $\arctan (8)$ .

$\frac{3 e^{5 (\frac{1}{8})} - 1 \cdot 3}{π - 2 \cdot 1.44644133}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Combine $5$ and $\frac{1}{8}$ .

$\frac{3 e^{\frac{5}{8}} - 1 \cdot 3}{π - 2 \cdot 1.44644133}$

Step 13.1.2

Multiply $- 1$ by $3$ .

$\frac{3 e^{\frac{5}{8}} - 3}{π - 2 \cdot 1.44644133}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Multiply $- 2$ by $1.44644133$ .

$\frac{3 e^{\frac{5}{8}} - 3}{π - 2.89288266}$

Step 13.2.2

Replace with decimal approximation.

$\frac{3 e^{\frac{5}{8}} - 3}{3.14159265 - 2.89288266}$

Step 13.2.3

Subtract $2.89288266$ from $3.14159265$ .

$\frac{3 e^{\frac{5}{8}} - 3}{0.24870998}$

Step 13.3

Replace $e$ with an approximation.

$\frac{3 \cdot {2.71828182}^{\frac{5}{8}} - 3}{0.24870998}$

Step 13.4

Divide $5$ by $8$ .

$\frac{3 \cdot {2.71828182}^{0.625} - 3}{0.24870998}$

Step 13.5

Raise $2.71828182$ to the power of $0.625$ .

$\frac{3 \cdot 1.86824595 - 3}{0.24870998}$

Step 13.6

Multiply $3$ by $1.86824595$ .

$\frac{5.60473787 - 3}{0.24870998}$

Step 13.7

Subtract $3$ from $5.60473787$ .

$\frac{2.60473787}{0.24870998}$

Step 13.8

Divide $2.60473787$ by $0.24870998$ .

$10.47299258$