Calculus Examples

$lim_{x \to 4} \frac{2 x^{2} + 5 x - 12}{x + 4}$

Step 1

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

$\frac{lim_{x \to 4} 2 x^{2} + 5 x - 12}{lim_{x \to 4} x + 4}$

Step 2

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

$\frac{lim_{x \to 4} 2 x^{2} + lim_{x \to 4} 5 x - lim_{x \to 4} 12}{lim_{x \to 4} x + 4}$

Step 3

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

$\frac{2 lim_{x \to 4} x^{2} + lim_{x \to 4} 5 x - lim_{x \to 4} 12}{lim_{x \to 4} x + 4}$

Step 4

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

$\frac{2 {(lim_{x \to 4} x)}^{2} + lim_{x \to 4} 5 x - lim_{x \to 4} 12}{lim_{x \to 4} x + 4}$

Step 5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 {(lim_{x \to 4} x)}^{2} + 5 lim_{x \to 4} x - lim_{x \to 4} 12}{lim_{x \to 4} x + 4}$

Step 6

Evaluate the limit of $12$ which is constant as $x$ approaches $4$ .

$\frac{2 {(lim_{x \to 4} x)}^{2} + 5 lim_{x \to 4} x - 1 \cdot 12}{lim_{x \to 4} x + 4}$

Step 7

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

$\frac{2 {(lim_{x \to 4} x)}^{2} + 5 lim_{x \to 4} x - 1 \cdot 12}{lim_{x \to 4} x + lim_{x \to 4} 4}$

Step 8

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

$\frac{2 {(lim_{x \to 4} x)}^{2} + 5 lim_{x \to 4} x - 1 \cdot 12}{lim_{x \to 4} x + 4}$

Step 9

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

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Step 9.1

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

$\frac{2 \cdot 4^{2} + 5 lim_{x \to 4} x - 1 \cdot 12}{lim_{x \to 4} x + 4}$

Step 9.2

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

$\frac{2 \cdot 4^{2} + 5 \cdot 4 - 1 \cdot 12}{lim_{x \to 4} x + 4}$

Step 9.3

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

$\frac{2 \cdot 4^{2} + 5 \cdot 4 - 1 \cdot 12}{4 + 4}$

Step 10

Simplify the answer.

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Step 10.1

Cancel the common factor of $2 \cdot 4^{2} + 5 \cdot 4 - 1 \cdot 12$ and $4 + 4$ .

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Step 10.1.1

Factor $- 1$ out of $2 \cdot 4^{2}$ .

$\frac{- (- 2 \cdot 4^{2}) + 5 \cdot 4 - 1 \cdot 12}{4 + 4}$

Step 10.1.2

Factor $- 1$ out of $5 \cdot 4$ .

$\frac{- (- 2 \cdot 4^{2}) - (- 5 \cdot 4) - 1 \cdot 12}{4 + 4}$

Step 10.1.3

Factor $- 1$ out of $- (- 2 \cdot 4^{2}) - (- 5 \cdot 4)$ .

$\frac{- (- 2 \cdot 4^{2} - 5 \cdot 4) - 1 \cdot 12}{4 + 4}$

Step 10.1.4

Factor $- 1$ out of $- 1 \cdot 12$ .

$\frac{- (- 2 \cdot 4^{2} - 5 \cdot 4) - 1 (12)}{4 + 4}$

Step 10.1.5

Factor $- 1$ out of $- (- 2 \cdot 4^{2} - 5 \cdot 4) - 1 (12)$ .

$\frac{- (- 2 \cdot 4^{2} - 5 \cdot 4 + 12)}{4 + 4}$

Step 10.1.6

Rewrite $- (- 2 \cdot 4^{2} - 5 \cdot 4 + 12)$ as $- 1 (- 2 \cdot 4^{2} - 5 \cdot 4 + 12)$ .

$\frac{- 1 (- 2 \cdot 4^{2} - 5 \cdot 4 + 12)}{4 + 4}$

Step 10.1.7

Factor $4$ out of $- 1 (- 2 \cdot 4^{2} - 5 \cdot 4 + 12)$ .

$\frac{4 (- 1 (- 2 \cdot 4 - 5 + 3))}{4 + 4}$

Step 10.1.8

Cancel the common factors.

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Step 10.1.8.1

Factor $4$ out of $4$ .

$\frac{4 (- 1 (- 2 \cdot 4 - 5 + 3))}{4 \cdot 1 + 4}$

Step 10.1.8.2

Factor $4$ out of $4$ .

$\frac{4 (- 1 (- 2 \cdot 4 - 5 + 3))}{4 \cdot 1 + 4 (1)}$

Step 10.1.8.3

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

$\frac{4 (- 1 (- 2 \cdot 4 - 5 + 3))}{4 \cdot (1 + 1)}$

Step 10.1.8.4

Cancel the common factor.

$\frac{4 (- 1 (- 2 \cdot 4 - 5 + 3))}{4 \cdot (1 + 1)}$

Step 10.1.8.5

Rewrite the expression.

$\frac{- 1 (- 2 \cdot 4 - 5 + 3)}{1 + 1}$

Step 10.2

Simplify the numerator.

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Step 10.2.1

Multiply $- 2$ by $4$ .

$\frac{- 1 (- 8 - 5 + 3)}{1 + 1}$

Step 10.2.2

Subtract $5$ from $- 8$ .

$\frac{- 1 (- 13 + 3)}{1 + 1}$

Step 10.2.3

Add $- 13$ and $3$ .

$\frac{- 1 \cdot - 10}{1 + 1}$

Step 10.3

Add $1$ and $1$ .

$\frac{- 1 \cdot - 10}{2}$

Step 10.4

Multiply $- 1$ by $- 10$ .

$\frac{10}{2}$

Step 10.5

Divide $10$ by $2$ .

$5$