Precalculus Examples

$lim_{x \to - 7} \frac{x + 5}{7 - x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 7$ .

$\frac{lim_{x \to - 7} x + 5}{lim_{x \to - 7} 7 - x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 7$ .

$\frac{lim_{x \to - 7} x + lim_{x \to - 7} 5}{lim_{x \to - 7} 7 - x}$

Step 3

Evaluate the limit of $5$ which is constant as $x$ approaches $- 7$ .

$\frac{lim_{x \to - 7} x + 5}{lim_{x \to - 7} 7 - x}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 7$ .

$\frac{lim_{x \to - 7} x + 5}{lim_{x \to - 7} 7 - lim_{x \to - 7} x}$

Step 5

Evaluate the limit of $7$ which is constant as $x$ approaches $- 7$ .

$\frac{lim_{x \to - 7} x + 5}{7 - lim_{x \to - 7} x}$

Step 6

Evaluate the limits by plugging in $- 7$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $- 7$ for $x$ .

$\frac{- 7 + 5}{7 - lim_{x \to - 7} x}$

Step 6.2

Evaluate the limit of $x$ by plugging in $- 7$ for $x$ .

$\frac{- 7 + 5}{7 + 7}$

Step 7

Simplify the answer.

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Step 7.1

Add $- 7$ and $5$ .

$\frac{- 2}{7 + 7}$

Step 7.2

Add $7$ and $7$ .

$\frac{- 2}{14}$

Step 7.3

Cancel the common factor of $- 2$ and $14$ .

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Step 7.3.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{14}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $2$ out of $14$ .

$\frac{2 \cdot - 1}{2 \cdot 7}$

Step 7.3.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 7}$

Step 7.3.2.3

Rewrite the expression.

$\frac{- 1}{7}$

Step 7.4

Move the negative in front of the fraction.

$- \frac{1}{7}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{7}$

Decimal Form:

$- 0.\overline{142857}$