Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Evaluate the limit of $2$ which is constant as $x$ approaches $- 8$ .

$\frac{lim_{x \to - 8} x - 1 \cdot 2}{lim_{x \to - 8} x^{2} + 1}$

Step 4

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

$\frac{lim_{x \to - 8} x - 1 \cdot 2}{lim_{x \to - 8} x^{2} + lim_{x \to - 8} 1}$

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $- 1$ by $2$ .

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

Step 8.1.2

Subtract $2$ from $- 8$ .

$\frac{- 10}{{(- 8)}^{2} + 1}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

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

$\frac{- 10}{64 + 1}$

Step 8.2.2

Add $64$ and $1$ .

$\frac{- 10}{65}$

Step 8.3

Cancel the common factor of $- 10$ and $65$ .

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Step 8.3.1

Factor $5$ out of $- 10$ .

$\frac{5 (- 2)}{65}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $5$ out of $65$ .

$\frac{5 \cdot - 2}{5 \cdot 13}$

Step 8.3.2.2

Cancel the common factor.

$\frac{5 \cdot - 2}{5 \cdot 13}$

Step 8.3.2.3

Rewrite the expression.

$\frac{- 2}{13}$

Step 8.4

Move the negative in front of the fraction.

$- \frac{2}{13}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{13}$

Decimal Form:

$- 0.\overline{153846}$