Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

$\frac{{(lim_{x \to - 3} x)}^{2}}{lim_{x \to - 3} x - 1 \cdot 3}$

Step 5

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

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Step 5.1

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

$\frac{{(- 3)}^{2}}{lim_{x \to - 3} x - 1 \cdot 3}$

Step 5.2

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

$\frac{{(- 3)}^{2}}{- 3 - 1 \cdot 3}$

Step 6

Simplify the answer.

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Step 6.1

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

$\frac{9}{- 3 - 1 \cdot 3}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $- 1$ by $3$ .

$\frac{9}{- 3 - 3}$

Step 6.2.2

Subtract $3$ from $- 3$ .

$\frac{9}{- 6}$

Step 6.3

Cancel the common factor of $9$ and $- 6$ .

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Step 6.3.1

Factor $3$ out of $9$ .

$\frac{3 (3)}{- 6}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $3$ out of $- 6$ .

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

Step 6.3.2.2

Cancel the common factor.

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

Step 6.3.2.3

Rewrite the expression.

$\frac{3}{- 2}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2}$

Decimal Form:

$- 1.5$