Calculus Examples

$lim_{x \to - 3} (6 - 3 x) (\frac{3}{5} x^{- 1})$

Step 1

Move the term $\frac{3}{5}$ outside of the limit because it is constant with respect to $x$ .

$\frac{3}{5} lim_{x \to - 3} (6 - 3 x) x^{- 1}$

Step 2

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

$\frac{3}{5} lim_{x \to - 3} 6 - 3 x \cdot lim_{x \to - 3} x^{- 1}$

Step 3

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

$\frac{3}{5} (lim_{x \to - 3} 6 - lim_{x \to - 3} 3 x) \cdot lim_{x \to - 3} x^{- 1}$

Step 4

Evaluate the limit of $6$ which is constant as $x$ approaches $- 3$ .

$\frac{3}{5} (6 - lim_{x \to - 3} 3 x) \cdot lim_{x \to - 3} x^{- 1}$

Step 5

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

$\frac{3}{5} (6 - 3 lim_{x \to - 3} x) \cdot lim_{x \to - 3} x^{- 1}$

Step 6

Move the exponent $- 1$ from $x^{- 1}$ outside the limit using the Limits Power Rule.

$\frac{3}{5} (6 - 3 lim_{x \to - 3} x) \cdot {(lim_{x \to - 3} x)}^{- 1}$

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 $x$ .

$\frac{3}{5} (6 - 3 \cdot - 3) \cdot {(lim_{x \to - 3} x)}^{- 1}$

Step 7.2

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

$\frac{3}{5} (6 - 3 \cdot - 3) \cdot {(- 3)}^{- 1}$

Step 8

Simplify the answer.

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Step 8.1

Multiply $- 3$ by $- 3$ .

$\frac{3}{5} (6 + 9) \cdot {(- 3)}^{- 1}$

Step 8.2

Add $6$ and $9$ .

$\frac{3}{5} \cdot 15 \cdot {(- 3)}^{- 1}$

Step 8.3

Cancel the common factor of $5$ .

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Step 8.3.1

Factor $5$ out of $15$ .

$\frac{3}{5} \cdot (5 (3)) \cdot {(- 3)}^{- 1}$

Step 8.3.2

Cancel the common factor.

$\frac{3}{5} \cdot (5 \cdot 3) \cdot {(- 3)}^{- 1}$

Step 8.3.3

Rewrite the expression.

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

Step 8.4

Multiply $3$ by $3$ .

$9 \cdot {(- 3)}^{- 1}$

Step 8.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8.6

Cancel the common factor of $3$ .

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Step 8.6.1

Factor $3$ out of $9$ .

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

Step 8.6.2

Factor $3$ out of $- 3$ .

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

Step 8.6.3

Cancel the common factor.

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

Step 8.6.4

Rewrite the expression.

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

Step 8.7

Combine $3$ and $\frac{1}{- 1}$ .

$\frac{3}{- 1}$

Step 8.8

Divide $3$ by $- 1$ .

$- 3$