Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $- 8$ .

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

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

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

Step 8

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

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

Step 9

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

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

Step 10

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

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

Step 10.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

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

Step 10.3

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

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

Step 10.4

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

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

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raise $- 8$ to the power of $2$ .

$\frac{2 \cdot 64 + 3 \cdot - 8 + 1}{{(- 8)}^{4} - {(- 8)}^{2}}$

Step 11.1.2

Multiply $2$ by $64$ .

$\frac{128 + 3 \cdot - 8 + 1}{{(- 8)}^{4} - {(- 8)}^{2}}$

Step 11.1.3

Multiply $3$ by $- 8$ .

$\frac{128 - 24 + 1}{{(- 8)}^{4} - {(- 8)}^{2}}$

Step 11.1.4

Subtract $24$ from $128$ .

$\frac{104 + 1}{{(- 8)}^{4} - {(- 8)}^{2}}$

Step 11.1.5

Add $104$ and $1$ .

$\frac{105}{{(- 8)}^{4} - {(- 8)}^{2}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Raise $- 8$ to the power of $4$ .

$\frac{105}{4096 - {(- 8)}^{2}}$

Step 11.2.2

Raise $- 8$ to the power of $2$ .

$\frac{105}{4096 - 1 \cdot 64}$

Step 11.2.3

Multiply $- 1$ by $64$ .

$\frac{105}{4096 - 64}$

Step 11.2.4

Subtract $64$ from $4096$ .

$\frac{105}{4032}$

Step 11.3

Cancel the common factor of $105$ and $4032$ .

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Step 11.3.1

Factor $21$ out of $105$ .

$\frac{21 (5)}{4032}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $21$ out of $4032$ .

$\frac{21 \cdot 5}{21 \cdot 192}$

Step 11.3.2.2

Cancel the common factor.

$\frac{21 \cdot 5}{21 \cdot 192}$

Step 11.3.2.3

Rewrite the expression.

$\frac{5}{192}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{192}$

Decimal Form:

$0.026041\overline{6}$