Calculus Examples

$lim_{x \to 4} \frac{f (x) - 5}{x - 2}$

Step 1

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

$\frac{lim_{x \to 4} f (x) - 5}{lim_{x \to 4} x - 2}$

Step 2

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

$\frac{lim_{x \to 4} f (x) - lim_{x \to 4} 5}{lim_{x \to 4} x - 2}$

Step 3

Evaluate the limit of $5$ which is constant as $x$ approaches $4$ .

$\frac{lim_{x \to 4} f (x) - 1 \cdot 5}{lim_{x \to 4} x - 2}$

Step 4

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

$\frac{lim_{x \to 4} f (x) - 1 \cdot 5}{lim_{x \to 4} x - lim_{x \to 4} 2}$

Step 5

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

$\frac{lim_{x \to 4} f (x) - 1 \cdot 5}{lim_{x \to 4} x - 1 \cdot 2}$

Step 6

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

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Step 6.1

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

$\frac{f (4) - 1 \cdot 5}{lim_{x \to 4} x - 1 \cdot 2}$

Step 6.2

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

$\frac{f (4) - 1 \cdot 5}{4 - 1 \cdot 2}$

Step 7

Simplify the answer.

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Step 7.1

Multiply $- 1$ by $5$ .

$\frac{f (4) - 5}{4 - 1 \cdot 2}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Multiply $- 1$ by $2$ .

$\frac{f (4) - 5}{4 - 2}$

Step 7.2.2

Subtract $2$ from $4$ .

$\frac{f (4) - 5}{2}$