Calculus Examples

$lim_{x \to - 3} \frac{4 x + 36}{3 - x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 3$ .

$\frac{lim_{x \to - 3} 4 x + 36}{lim_{x \to - 3} 3 - x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{lim_{x \to - 3} 4 x + lim_{x \to - 3} 36}{lim_{x \to - 3} 3 - x}$

Step 3

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

$\frac{4 lim_{x \to - 3} x + lim_{x \to - 3} 36}{lim_{x \to - 3} 3 - x}$

Step 4

Evaluate the limit of $36$ which is constant as $x$ approaches $- 3$ .

$\frac{4 lim_{x \to - 3} x + 36}{lim_{x \to - 3} 3 - x}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{4 lim_{x \to - 3} x + 36}{lim_{x \to - 3} 3 - lim_{x \to - 3} x}$

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $- 3$ .

$\frac{4 lim_{x \to - 3} x + 36}{3 - lim_{x \to - 3} x}$

Step 7

Evaluate the limits by plugging in $- 3$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{4 \cdot - 3 + 36}{3 - lim_{x \to - 3} x}$

Step 7.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{4 \cdot - 3 + 36}{3 + 3}$

Step 8

Simplify the answer.

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Step 8.1

Cancel the common factor of $4 \cdot - 3 + 36$ and $3 + 3$ .

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Step 8.1.1

Factor $3$ out of $4 \cdot - 3$ .

$\frac{3 (4 \cdot - 1) + 36}{3 + 3}$

Step 8.1.2

Factor $3$ out of $36$ .

$\frac{3 (4 \cdot - 1) + 3 \cdot 12}{3 + 3}$

Step 8.1.3

Factor $3$ out of $3 (4 \cdot - 1) + 3 (12)$ .

$\frac{3 (4 \cdot - 1 + 12)}{3 + 3}$

Step 8.1.4

Cancel the common factors.

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Step 8.1.4.1

Factor $3$ out of $3$ .

$\frac{3 (4 \cdot - 1 + 12)}{3 \cdot 1 + 3}$

Step 8.1.4.2

Factor $3$ out of $3$ .

$\frac{3 (4 \cdot - 1 + 12)}{3 \cdot 1 + 3 (1)}$

Step 8.1.4.3

Factor $3$ out of $3 \cdot 1 + 3 (1)$ .

$\frac{3 (4 \cdot - 1 + 12)}{3 \cdot (1 + 1)}$

Step 8.1.4.4

Cancel the common factor.

$\frac{3 (4 \cdot - 1 + 12)}{3 \cdot (1 + 1)}$

Step 8.1.4.5

Rewrite the expression.

$\frac{4 \cdot - 1 + 12}{1 + 1}$

Step 8.2

Simplify the numerator.

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Step 8.2.1

Multiply $4$ by $- 1$ .

$\frac{- 4 + 12}{1 + 1}$

Step 8.2.2

Add $- 4$ and $12$ .

$\frac{8}{1 + 1}$

Step 8.3

Add $1$ and $1$ .

$\frac{8}{2}$

Step 8.4

Divide $8$ by $2$ .

$4$