Calculus Examples

$lim_{x \to 4} \frac{3 x}{(x - 2) (x + 2)}$

Step 1

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

$3 lim_{x \to 4} \frac{x}{(x - 2) (x + 2)}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$3 \frac{lim_{x \to 4} x}{lim_{x \to 4} (x - 2) (x + 2)}$

Step 3

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $4$ .

$3 \frac{lim_{x \to 4} x}{lim_{x \to 4} x - 2 \cdot lim_{x \to 4} x + 2}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$3 \frac{lim_{x \to 4} x}{(lim_{x \to 4} x - lim_{x \to 4} 2) \cdot lim_{x \to 4} x + 2}$

Step 5

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

$3 \frac{lim_{x \to 4} x}{(lim_{x \to 4} x - 1 \cdot 2) \cdot lim_{x \to 4} x + 2}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$3 \frac{lim_{x \to 4} x}{(lim_{x \to 4} x - 1 \cdot 2) \cdot (lim_{x \to 4} x + lim_{x \to 4} 2)}$

Step 7

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

$3 \frac{lim_{x \to 4} x}{(lim_{x \to 4} x - 1 \cdot 2) \cdot (lim_{x \to 4} x + 2)}$

Step 8

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$3 \frac{4}{(lim_{x \to 4} x - 1 \cdot 2) \cdot (lim_{x \to 4} x + 2)}$

Step 8.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$3 \frac{4}{(4 - 1 \cdot 2) \cdot (lim_{x \to 4} x + 2)}$

Step 8.3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$3 \frac{4}{(4 - 1 \cdot 2) \cdot (4 + 2)}$

Step 9

Simplify the answer.

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Step 9.1

Cancel the common factor of $4$ and $4 + 2$ .

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Step 9.1.1

Factor $2$ out of $4$ .

$3 \frac{2 \cdot 2}{(4 - 1 \cdot 2) \cdot (4 + 2)}$

Step 9.1.2

Cancel the common factors.

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Step 9.1.2.1

Factor $2$ out of $(4 - 1 \cdot 2) \cdot (4 + 2)$ .

$3 \frac{2 \cdot 2}{2 ((2 - 1) \cdot (4 + 2))}$

Step 9.1.2.2

Cancel the common factor.

$3 \frac{2 \cdot 2}{2 ((2 - 1) \cdot (4 + 2))}$

Step 9.1.2.3

Rewrite the expression.

$3 \frac{2}{(2 - 1) \cdot (4 + 2)}$

Step 9.2

Cancel the common factor of $2$ and $4 + 2$ .

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Step 9.2.1

Factor $2$ out of $2$ .

$3 \frac{2 \cdot 1}{(2 - 1) \cdot (4 + 2)}$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $2$ out of $(2 - 1) \cdot (4 + 2)$ .

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

Step 9.2.2.2

Cancel the common factor.

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

Step 9.2.2.3

Rewrite the expression.

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

Step 9.3

Simplify the denominator.

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Step 9.3.1

Subtract $1$ from $2$ .

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

Step 9.3.2

Multiply $2 + 1$ by $1$ .

$3 \frac{1}{2 + 1}$

Step 9.3.3

Add $2$ and $1$ .

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

Step 9.4

Cancel the common factor of $3$ .

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Step 9.4.1

Cancel the common factor.

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

Step 9.4.2

Rewrite the expression.

$1$