Calculus Examples

$lim_{x \to - 3} \frac{x^{3} + 5}{2 - 3 x}$

Step 1

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

$\frac{lim_{x \to - 3} x^{3} + 5}{lim_{x \to - 3} 2 - 3 x}$

Step 2

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

$\frac{lim_{x \to - 3} x^{3} + lim_{x \to - 3} 5}{lim_{x \to - 3} 2 - 3 x}$

Step 3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to - 3} x)}^{3} + lim_{x \to - 3} 5}{lim_{x \to - 3} 2 - 3 x}$

Step 4

Evaluate the limit of $5$ which is constant as $x$ approaches $- 3$ .

$\frac{{(lim_{x \to - 3} x)}^{3} + 5}{lim_{x \to - 3} 2 - 3 x}$

Step 5

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

$\frac{{(lim_{x \to - 3} x)}^{3} + 5}{lim_{x \to - 3} 2 - lim_{x \to - 3} 3 x}$

Step 6

Evaluate the limit of $2$ which is constant as $x$ approaches $- 3$ .

$\frac{{(lim_{x \to - 3} x)}^{3} + 5}{2 - lim_{x \to - 3} 3 x}$

Step 7

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

$\frac{{(lim_{x \to - 3} x)}^{3} + 5}{2 - 3 lim_{x \to - 3} x}$

Step 8

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

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Step 8.1

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

$\frac{{(- 3)}^{3} + 5}{2 - 3 lim_{x \to - 3} x}$

Step 8.2

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

$\frac{{(- 3)}^{3} + 5}{2 - 3 \cdot - 3}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{- 27 + 5}{2 - 3 \cdot - 3}$

Step 9.1.2

Add $- 27$ and $5$ .

$\frac{- 22}{2 - 3 \cdot - 3}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $- 3$ by $- 3$ .

$\frac{- 22}{2 + 9}$

Step 9.2.2

Add $2$ and $9$ .

$\frac{- 22}{11}$

Step 9.3

Divide $- 22$ by $11$ .

$- 2$