Calculus Examples

$\frac{lim_{x \to 8} \frac{1}{x^{2}} + 2 \arctan (\frac{1}{x})}{\frac{1}{x}}$

Step 1

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

$\frac{lim_{x \to 8} \frac{1}{x^{2}} + lim_{x \to 8} 2 \arctan (\frac{1}{x})}{\frac{1}{x}}$

Step 2

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

$\frac{\frac{lim_{x \to 8} 1}{lim_{x \to 8} x^{2}} + lim_{x \to 8} 2 \arctan (\frac{1}{x})}{\frac{1}{x}}$

Step 3

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

$\frac{\frac{1}{lim_{x \to 8} x^{2}} + lim_{x \to 8} 2 \arctan (\frac{1}{x})}{\frac{1}{x}}$

Step 4

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

$\frac{\frac{1}{{(lim_{x \to 8} x)}^{2}} + lim_{x \to 8} 2 \arctan (\frac{1}{x})}{\frac{1}{x}}$

Step 5

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

$\frac{\frac{1}{{(lim_{x \to 8} x)}^{2}} + 2 lim_{x \to 8} \arctan (\frac{1}{x})}{\frac{1}{x}}$

Step 6

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

$\frac{\frac{1}{{(lim_{x \to 8} x)}^{2}} + 2 (\frac{1}{8})}{\frac{1}{x}}$

Step 7

Substitute $t$ for $\frac{1}{x}$ and let $t$ approach $\frac{1}{8}$ since $lim_{x \to 8} \frac{1}{x} = \frac{1}{8}$ .

$\frac{\frac{1}{{(lim_{x \to 8} x)}^{2}} + 2 lim_{t \to \frac{1}{8}} \arctan (t)}{\frac{1}{x}}$

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 $x$ .

$\frac{\frac{1}{8^{2}} + 2 lim_{t \to \frac{1}{8}} \arctan (t)}{\frac{1}{x}}$

Step 8.2

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

$\frac{\frac{1}{8^{2}} + 2 \arctan (\frac{1}{8})}{\frac{1}{x}}$

Step 8.3

Evaluate $\arctan (\frac{1}{8})$ .

$\frac{\frac{1}{8^{2}} + 2 \cdot 0.12435499}{\frac{1}{x}}$

Step 9

Simplify the answer.

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Step 9.1

Multiply the numerator by the reciprocal of the denominator.

$(\frac{1}{8^{2}} + 2 \cdot 0.12435499) x$

Step 9.2

Simplify each term.

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Step 9.2.1

Raise $8$ to the power of $2$ .

$(\frac{1}{64} + 2 \cdot 0.12435499) x$

Step 9.2.2

Multiply $2$ by $0.12435499$ .

$(\frac{1}{64} + 0.24870998) x$

Step 9.3

To write $0.24870998$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$(\frac{1}{64} + 0.24870998 \cdot \frac{64}{64}) x$

Step 9.4

Combine $0.24870998$ and $\frac{64}{64}$ .

$(\frac{1}{64} + \frac{0.24870998 \cdot 64}{64}) x$

Step 9.5

Combine the numerators over the common denominator.

$\frac{1 + 0.24870998 \cdot 64}{64} x$

Step 9.6

Simplify the numerator.

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Step 9.6.1

Multiply $0.24870998$ by $64$ .

$\frac{1 + 15.9174393}{64} x$

Step 9.6.2

Add $1$ and $15.9174393$ .

$\frac{16.9174393}{64} x$

Step 9.7

Divide $16.9174393$ by $64$ .

$0.26433498 x$