Calculus Examples

$lim_{x \to 7} \frac{- 4 x}{x + 7}$

Step 1

Move the term $- 4$ outside of the limit because it is constant with respect to $x$ .

$- 4 lim_{x \to 7} \frac{x}{x + 7}$

Step 2

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

$- 4 \frac{lim_{x \to 7} x}{lim_{x \to 7} x + 7}$

Step 3

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

$- 4 \frac{lim_{x \to 7} x}{lim_{x \to 7} x + lim_{x \to 7} 7}$

Step 4

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

$- 4 \frac{lim_{x \to 7} x}{lim_{x \to 7} x + 7}$

Step 5

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

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Step 5.1

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

$- 4 \frac{7}{lim_{x \to 7} x + 7}$

Step 5.2

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

$- 4 \frac{7}{7 + 7}$

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $7$ and $7 + 7$ .

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Step 6.1.1

Factor $7$ out of $7$ .

$- 4 \frac{7 \cdot 1}{7 + 7}$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $7$ out of $7$ .

$- 4 \frac{7 \cdot 1}{7 (1) + 7}$

Step 6.1.2.2

Factor $7$ out of $7$ .

$- 4 \frac{7 \cdot 1}{7 \cdot 1 + 7 \cdot 1}$

Step 6.1.2.3

Factor $7$ out of $7 \cdot 1 + 7 \cdot 1$ .

$- 4 \frac{7 \cdot 1}{7 \cdot (1 + 1)}$

Step 6.1.2.4

Cancel the common factor.

$- 4 \frac{7 \cdot 1}{7 \cdot (1 + 1)}$

Step 6.1.2.5

Rewrite the expression.

$- 4 \frac{1}{1 + 1}$

Step 6.2

Add $1$ and $1$ .

$- 4 (\frac{1}{2})$

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Factor $2$ out of $- 4$ .

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

Step 6.3.2

Cancel the common factor.

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

Step 6.3.3

Rewrite the expression.

$- 2$