Calculus Examples

$lim_{x \to - 3} \frac{3 x^{2} - x}{x + 4}$

Step 1

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

$\frac{lim_{x \to - 3} 3 x^{2} - x}{lim_{x \to - 3} x + 4}$

Step 2

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

$\frac{lim_{x \to - 3} 3 x^{2} - lim_{x \to - 3} x}{lim_{x \to - 3} x + 4}$

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to - 3} x^{2} - lim_{x \to - 3} x}{lim_{x \to - 3} x + 4}$

Step 4

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

$\frac{3 {(lim_{x \to - 3} x)}^{2} - lim_{x \to - 3} x}{lim_{x \to - 3} x + 4}$

Step 5

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

$\frac{3 {(lim_{x \to - 3} x)}^{2} - lim_{x \to - 3} x}{lim_{x \to - 3} x + lim_{x \to - 3} 4}$

Step 6

Evaluate the limit of $4$ which is constant as $x$ approaches $- 3$ .

$\frac{3 {(lim_{x \to - 3} x)}^{2} - lim_{x \to - 3} x}{lim_{x \to - 3} x + 4}$

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 $x$ .

$\frac{3 {(- 3)}^{2} - lim_{x \to - 3} x}{lim_{x \to - 3} x + 4}$

Step 7.2

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

$\frac{3 {(- 3)}^{2} + 3}{lim_{x \to - 3} x + 4}$

Step 7.3

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

$\frac{3 {(- 3)}^{2} + 3}{- 3 + 4}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 3$ to the power of $2$ .

$\frac{3 \cdot 9 + 3}{- 3 + 4}$

Step 8.1.2

Multiply $3$ by $9$ .

$\frac{27 + 3}{- 3 + 4}$

Step 8.1.3

Add $27$ and $3$ .

$\frac{30}{- 3 + 4}$

Step 8.2

Add $- 3$ and $4$ .

$\frac{30}{1}$

Step 8.3

Divide $30$ by $1$ .

$30$