Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

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 $x$ .

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

Step 7.2

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

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

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 $2$ .

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

Step 8.1.2

Multiply $- 1$ by $1$ .

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

Step 8.1.3

Subtract $1$ from $9$ .

$\frac{8}{3 + 3}$

Step 8.2

Add $3$ and $3$ .

$\frac{8}{6}$

Step 8.3

Cancel the common factor of $8$ and $6$ .

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Step 8.3.1

Factor $2$ out of $8$ .

$\frac{2 (4)}{6}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 4}{2 \cdot 3}$

Step 8.3.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot 3}$

Step 8.3.2.3

Rewrite the expression.

$\frac{4}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{3}$

Decimal Form:

$1.\overline{3}$