Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

$- 3 (\frac{1}{9})$

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Factor $3$ out of $- 3$ .

$3 (- 1) \frac{1}{9}$

Step 3.2.2

Factor $3$ out of $9$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

$- \frac{1}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{3}$

Decimal Form:

$- 0.\overline{3}$