Calculus Examples

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

Step 1

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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 $x$ .

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

Step 5.2

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

$2 \frac{- 3}{- 3 - 1 \cdot 3}$

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $- 3$ and $- 3 - 1 \cdot 3$ .

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Step 6.1.1

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 6.1.2

Factor $- 1$ out of $- 1 \cdot 3$ .

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

Step 6.1.3

Factor $- 1$ out of $- 1 (3) - 1 (3)$ .

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

Step 6.1.4

Factor $3$ out of $- 3$ .

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

Step 6.1.5

Cancel the common factors.

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Step 6.1.5.1

Factor $3$ out of $- 1 (3 + 3)$ .

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

Step 6.1.5.2

Cancel the common factor.

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

Step 6.1.5.3

Rewrite the expression.

$2 \frac{- 1}{- 1 (1 + 1)}$

Step 6.2

Dividing two negative values results in a positive value.

$2 \frac{1}{1 + 1}$

Step 6.3

Add $1$ and $1$ .

$2 (\frac{1}{2})$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$2 (\frac{1}{2})$

Step 6.4.2

Rewrite the expression.

$1$