Calculus Examples

$lim_{x \to 4} \frac{5 x^{2} - 22 x + 8}{2 x^{3} - 7 x - 4}$

Step 1

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

$\frac{lim_{x \to 4} 5 x^{2} - 22 x + 8}{lim_{x \to 4} 2 x^{3} - 7 x - 4}$

Step 2

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

$\frac{lim_{x \to 4} 5 x^{2} - lim_{x \to 4} 22 x + lim_{x \to 4} 8}{lim_{x \to 4} 2 x^{3} - 7 x - 4}$

Step 3

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

$\frac{5 lim_{x \to 4} x^{2} - lim_{x \to 4} 22 x + lim_{x \to 4} 8}{lim_{x \to 4} 2 x^{3} - 7 x - 4}$

Step 4

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

$\frac{5 {(lim_{x \to 4} x)}^{2} - lim_{x \to 4} 22 x + lim_{x \to 4} 8}{lim_{x \to 4} 2 x^{3} - 7 x - 4}$

Step 5

Move the term $22$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 {(lim_{x \to 4} x)}^{2} - 22 lim_{x \to 4} x + lim_{x \to 4} 8}{lim_{x \to 4} 2 x^{3} - 7 x - 4}$

Step 6

Evaluate the limit of $8$ which is constant as $x$ approaches $4$ .

$\frac{5 {(lim_{x \to 4} x)}^{2} - 22 lim_{x \to 4} x + 8}{lim_{x \to 4} 2 x^{3} - 7 x - 4}$

Step 7

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

$\frac{5 {(lim_{x \to 4} x)}^{2} - 22 lim_{x \to 4} x + 8}{lim_{x \to 4} 2 x^{3} - lim_{x \to 4} 7 x - lim_{x \to 4} 4}$

Step 8

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

$\frac{5 {(lim_{x \to 4} x)}^{2} - 22 lim_{x \to 4} x + 8}{2 lim_{x \to 4} x^{3} - lim_{x \to 4} 7 x - lim_{x \to 4} 4}$

Step 9

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{5 {(lim_{x \to 4} x)}^{2} - 22 lim_{x \to 4} x + 8}{2 {(lim_{x \to 4} x)}^{3} - lim_{x \to 4} 7 x - lim_{x \to 4} 4}$

Step 10

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 {(lim_{x \to 4} x)}^{2} - 22 lim_{x \to 4} x + 8}{2 {(lim_{x \to 4} x)}^{3} - 7 lim_{x \to 4} x - lim_{x \to 4} 4}$

Step 11

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

$\frac{5 {(lim_{x \to 4} x)}^{2} - 22 lim_{x \to 4} x + 8}{2 {(lim_{x \to 4} x)}^{3} - 7 lim_{x \to 4} x - 1 \cdot 4}$

Step 12

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

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Step 12.1

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

$\frac{5 \cdot 4^{2} - 22 lim_{x \to 4} x + 8}{2 {(lim_{x \to 4} x)}^{3} - 7 lim_{x \to 4} x - 1 \cdot 4}$

Step 12.2

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

$\frac{5 \cdot 4^{2} - 22 \cdot 4 + 8}{2 {(lim_{x \to 4} x)}^{3} - 7 lim_{x \to 4} x - 1 \cdot 4}$

Step 12.3

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

$\frac{5 \cdot 4^{2} - 22 \cdot 4 + 8}{2 \cdot 4^{3} - 7 lim_{x \to 4} x - 1 \cdot 4}$

Step 12.4

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

$\frac{5 \cdot 4^{2} - 22 \cdot 4 + 8}{2 \cdot 4^{3} - 7 \cdot 4 - 1 \cdot 4}$

Step 13

Simplify the answer.

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Step 13.1

Cancel the common factor of $5 \cdot 4^{2} - 22 \cdot 4 + 8$ and $2 \cdot 4^{3} - 7 \cdot 4 - 1 \cdot 4$ .

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Step 13.1.1

Factor $- 1$ out of $2 \cdot 4^{3}$ .

$\frac{5 \cdot 4^{2} - 22 \cdot 4 + 8}{- (- 2 \cdot 4^{3}) - 7 \cdot 4 - 1 \cdot 4}$

Step 13.1.2

Factor $- 1$ out of $- 7 \cdot 4$ .

$\frac{5 \cdot 4^{2} - 22 \cdot 4 + 8}{- (- 2 \cdot 4^{3}) - (7 \cdot 4) - 1 \cdot 4}$

Step 13.1.3

Factor $- 1$ out of $- (- 2 \cdot 4^{3}) - (7 \cdot 4)$ .

$\frac{5 \cdot 4^{2} - 22 \cdot 4 + 8}{- (- 2 \cdot 4^{3} + 7 \cdot 4) - 1 \cdot 4}$

Step 13.1.4

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

$\frac{5 \cdot 4^{2} - 22 \cdot 4 + 8}{- (- 2 \cdot 4^{3} + 7 \cdot 4) - 1 (4)}$

Step 13.1.5

Factor $- 1$ out of $- (- 2 \cdot 4^{3} + 7 \cdot 4) - 1 (4)$ .

$\frac{5 \cdot 4^{2} - 22 \cdot 4 + 8}{- (- 2 \cdot 4^{3} + 7 \cdot 4 + 4)}$

Step 13.1.6

Rewrite $- (- 2 \cdot 4^{3} + 7 \cdot 4 + 4)$ as $- 1 (- 2 \cdot 4^{3} + 7 \cdot 4 + 4)$ .

$\frac{5 \cdot 4^{2} - 22 \cdot 4 + 8}{- 1 (- 2 \cdot 4^{3} + 7 \cdot 4 + 4)}$

Step 13.1.7

Factor $4$ out of $5 \cdot 4^{2}$ .

$\frac{4 (5 \cdot 4) - 22 \cdot 4 + 8}{- 1 (- 2 \cdot 4^{3} + 7 \cdot 4 + 4)}$

Step 13.1.8

Factor $4$ out of $- 22 \cdot 4$ .

$\frac{4 (5 \cdot 4) + 4 \cdot - 22 + 8}{- 1 (- 2 \cdot 4^{3} + 7 \cdot 4 + 4)}$

Step 13.1.9

Factor $4$ out of $4 (5 \cdot 4) + 4 \cdot - 22$ .

$\frac{4 (5 \cdot 4 - 22) + 8}{- 1 (- 2 \cdot 4^{3} + 7 \cdot 4 + 4)}$

Step 13.1.10

Factor $4$ out of $8$ .

$\frac{4 (5 \cdot 4 - 22) + 4 \cdot 2}{- 1 (- 2 \cdot 4^{3} + 7 \cdot 4 + 4)}$

Step 13.1.11

Factor $4$ out of $4 (5 \cdot 4 - 22) + 4 (2)$ .

$\frac{4 (5 \cdot 4 - 22 + 2)}{- 1 (- 2 \cdot 4^{3} + 7 \cdot 4 + 4)}$

Step 13.1.12

Cancel the common factors.

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Step 13.1.12.1

Factor $4$ out of $- 1 (- 2 \cdot 4^{3} + 7 \cdot 4 + 4)$ .

$\frac{4 (5 \cdot 4 - 22 + 2)}{4 (- 1 (- 2 \cdot 4^{2} + 7 + 1))}$

Step 13.1.12.2

Cancel the common factor.

$\frac{4 (5 \cdot 4 - 22 + 2)}{4 (- 1 (- 2 \cdot 4^{2} + 7 + 1))}$

Step 13.1.12.3

Rewrite the expression.

$\frac{5 \cdot 4 - 22 + 2}{- 1 (- 2 \cdot 4^{2} + 7 + 1)}$

Step 13.2

Simplify the numerator.

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Step 13.2.1

Multiply $5$ by $4$ .

$\frac{20 - 22 + 2}{- 1 (- 2 \cdot 4^{2} + 7 + 1)}$

Step 13.2.2

Subtract $22$ from $20$ .

$\frac{- 2 + 2}{- 1 (- 2 \cdot 4^{2} + 7 + 1)}$

Step 13.2.3

Add $- 2$ and $2$ .

$\frac{0}{- 1 (- 2 \cdot 4^{2} + 7 + 1)}$

Step 13.3

Simplify the denominator.

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Step 13.3.1

Raise $4$ to the power of $2$ .

$\frac{0}{- 1 (- 2 \cdot 16 + 7 + 1)}$

Step 13.3.2

Multiply $- 2$ by $16$ .

$\frac{0}{- 1 (- 32 + 7 + 1)}$

Step 13.3.3

Add $- 32$ and $7$ .

$\frac{0}{- 1 (- 25 + 1)}$

Step 13.3.4

Add $- 25$ and $1$ .

$\frac{0}{- 1 \cdot - 24}$

Step 13.4

Multiply $- 1$ by $- 24$ .

$\frac{0}{24}$

Step 13.5

Divide $0$ by $24$ .

$0$