Calculus Examples

$lim_{x \to 2} \frac{{(f (x))}^{2} + f (x) - 12}{f (x) + 4}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} {(f (x))}^{2} + f (x) - 12}{lim_{x \to 2} f (x) + 4}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} {(f (x))}^{2} + lim_{x \to 2} f (x) - lim_{x \to 2} 12}{lim_{x \to 2} f (x) + 4}$

Step 3

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

$\frac{{(lim_{x \to 2} f (x))}^{2} + lim_{x \to 2} f (x) - lim_{x \to 2} 12}{lim_{x \to 2} f (x) + 4}$

Step 4

Evaluate the limit of $12$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} f (x))}^{2} + lim_{x \to 2} f (x) - 1 \cdot 12}{lim_{x \to 2} f (x) + 4}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} f (x))}^{2} + lim_{x \to 2} f (x) - 1 \cdot 12}{lim_{x \to 2} f (x) + lim_{x \to 2} 4}$

Step 6

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

$\frac{{(lim_{x \to 2} f (x))}^{2} + lim_{x \to 2} f (x) - 1 \cdot 12}{lim_{x \to 2} f (x) + 4}$

Step 7

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $f (x)$ by plugging in $2$ for $x$ .

$\frac{{f (2)}^{2} + lim_{x \to 2} f (x) - 1 \cdot 12}{lim_{x \to 2} f (x) + 4}$

Step 7.2

Evaluate the limit of $f (x)$ by plugging in $2$ for $x$ .

$\frac{{f (2)}^{2} + f (2) - 1 \cdot 12}{lim_{x \to 2} f (x) + 4}$

Step 7.3

Evaluate the limit of $f (x)$ by plugging in $2$ for $x$ .

$\frac{{f (2)}^{2} + f (2) - 1 \cdot 12}{f (2) + 4}$

Step 8

Multiply $- 1$ by $12$ .

$\frac{{f (2)}^{2} + f (2) - 12}{f (2) + 4}$