Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

$\frac{4 lim_{x \to - 3} \frac{1}{x} + lim_{x \to - 3} \frac{4}{3}}{x - 3}$

Step 1.3

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

$\frac{4 \frac{lim_{x \to - 3} 1}{lim_{x \to - 3} x} + lim_{x \to - 3} \frac{4}{3}}{x - 3}$

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $- 3$ .

$\frac{4 \frac{1}{lim_{x \to - 3} x} + lim_{x \to - 3} \frac{4}{3}}{x - 3}$

Step 1.5

Evaluate the limit of $\frac{4}{3}$ which is constant as $x$ approaches $- 3$ .

$\frac{4 \frac{1}{lim_{x \to - 3} x} + \frac{4}{3}}{x - 3}$

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 3.1.1

Multiply $\frac{4 (\frac{1}{- 3}) + \frac{4}{3}}{x - 3}$ by $\frac{3}{3}$ .

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $3$ by $4$ .

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

Step 3.4.2

Cancel the common factor of $3$ .

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Step 3.4.2.1

Factor $3$ out of $12$ .

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

Step 3.4.2.2

Factor $3$ out of $- 3$ .

$\frac{3 \cdot 4 \frac{1}{3 \cdot - 1} + 4}{3 x + 3 \cdot - 3}$

Step 3.4.2.3

Cancel the common factor.

$\frac{3 \cdot 4 \frac{1}{3 \cdot - 1} + 4}{3 x + 3 \cdot - 3}$

Step 3.4.2.4

Rewrite the expression.

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

Step 3.4.3

Combine $4$ and $\frac{1}{- 1}$ .

$\frac{\frac{4}{- 1} + 4}{3 x + 3 \cdot - 3}$

Step 3.4.4

Divide $4$ by $- 1$ .

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

Step 3.4.5

Add $- 4$ and $4$ .

$\frac{0}{3 x + 3 \cdot - 3}$

Step 3.5

Factor $3$ out of $3 x + 3 \cdot - 3$ .

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Step 3.5.1

Factor $3$ out of $3 x$ .

$\frac{0}{3 (x) + 3 \cdot - 3}$

Step 3.5.2

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

$\frac{0}{3 (x) + 3 (- 3)}$

Step 3.5.3

Factor $3$ out of $3 (x) + 3 (- 3)$ .

$\frac{0}{3 (x - 3)}$

Step 3.6

Cancel the common factor of $0$ and $3$ .

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Step 3.6.1

Factor $3$ out of $0$ .

$\frac{3 (0)}{3 (x - 3)}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Cancel the common factor.

$\frac{3 \cdot 0}{3 (x - 3)}$

Step 3.6.2.2

Rewrite the expression.

$\frac{0}{x - 3}$

Step 3.7

Divide $0$ by $x - 3$ .

$0$