Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 2

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

$\tan (π \frac{1}{8^{2} + 1})$

Step 3

Simplify the answer.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

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

$\tan (π \frac{1}{64 + 1})$

Step 3.1.2

Add $64$ and $1$ .

$\tan (π \frac{1}{65})$

Step 3.2

Combine $π$ and $\frac{1}{65}$ .

$\tan (\frac{π}{65})$

Step 3.3

Evaluate $\tan (\frac{π}{65})$ .

$0.04836986$