Calculus Examples

$lim_{x \to 8} \frac{69 x^{2} + 35 - 83 x^{4}}{55 x^{4} - 91 x^{3} - 87}$

Step 1

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

$\frac{lim_{x \to 8} 69 x^{2} + 35 - 83 x^{4}}{lim_{x \to 8} 55 x^{4} - 91 x^{3} - 87}$

Step 2

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

$\frac{lim_{x \to 8} 69 x^{2} + lim_{x \to 8} 35 - lim_{x \to 8} 83 x^{4}}{lim_{x \to 8} 55 x^{4} - 91 x^{3} - 87}$

Step 3

Move the term $69$ outside of the limit because it is constant with respect to $x$ .

$\frac{69 lim_{x \to 8} x^{2} + lim_{x \to 8} 35 - lim_{x \to 8} 83 x^{4}}{lim_{x \to 8} 55 x^{4} - 91 x^{3} - 87}$

Step 4

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

$\frac{69 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 35 - lim_{x \to 8} 83 x^{4}}{lim_{x \to 8} 55 x^{4} - 91 x^{3} - 87}$

Step 5

Evaluate the limit of $35$ which is constant as $x$ approaches $8$ .

$\frac{69 {(lim_{x \to 8} x)}^{2} + 35 - lim_{x \to 8} 83 x^{4}}{lim_{x \to 8} 55 x^{4} - 91 x^{3} - 87}$

Step 6

Move the term $83$ outside of the limit because it is constant with respect to $x$ .

$\frac{69 {(lim_{x \to 8} x)}^{2} + 35 - 83 lim_{x \to 8} x^{4}}{lim_{x \to 8} 55 x^{4} - 91 x^{3} - 87}$

Step 7

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

$\frac{69 {(lim_{x \to 8} x)}^{2} + 35 - 83 {(lim_{x \to 8} x)}^{4}}{lim_{x \to 8} 55 x^{4} - 91 x^{3} - 87}$

Step 8

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

$\frac{69 {(lim_{x \to 8} x)}^{2} + 35 - 83 {(lim_{x \to 8} x)}^{4}}{lim_{x \to 8} 55 x^{4} - lim_{x \to 8} 91 x^{3} - lim_{x \to 8} 87}$

Step 9

Move the term $55$ outside of the limit because it is constant with respect to $x$ .

$\frac{69 {(lim_{x \to 8} x)}^{2} + 35 - 83 {(lim_{x \to 8} x)}^{4}}{55 lim_{x \to 8} x^{4} - lim_{x \to 8} 91 x^{3} - lim_{x \to 8} 87}$

Step 10

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

$\frac{69 {(lim_{x \to 8} x)}^{2} + 35 - 83 {(lim_{x \to 8} x)}^{4}}{55 {(lim_{x \to 8} x)}^{4} - lim_{x \to 8} 91 x^{3} - lim_{x \to 8} 87}$

Step 11

Move the term $91$ outside of the limit because it is constant with respect to $x$ .

$\frac{69 {(lim_{x \to 8} x)}^{2} + 35 - 83 {(lim_{x \to 8} x)}^{4}}{55 {(lim_{x \to 8} x)}^{4} - 91 lim_{x \to 8} x^{3} - lim_{x \to 8} 87}$

Step 12

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

$\frac{69 {(lim_{x \to 8} x)}^{2} + 35 - 83 {(lim_{x \to 8} x)}^{4}}{55 {(lim_{x \to 8} x)}^{4} - 91 {(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 87}$

Step 13

Evaluate the limit of $87$ which is constant as $x$ approaches $8$ .

$\frac{69 {(lim_{x \to 8} x)}^{2} + 35 - 83 {(lim_{x \to 8} x)}^{4}}{55 {(lim_{x \to 8} x)}^{4} - 91 {(lim_{x \to 8} x)}^{3} - 1 \cdot 87}$

Step 14

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

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Step 14.1

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

$\frac{69 \cdot 8^{2} + 35 - 83 {(lim_{x \to 8} x)}^{4}}{55 {(lim_{x \to 8} x)}^{4} - 91 {(lim_{x \to 8} x)}^{3} - 1 \cdot 87}$

Step 14.2

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

$\frac{69 \cdot 8^{2} + 35 - 83 \cdot 8^{4}}{55 {(lim_{x \to 8} x)}^{4} - 91 {(lim_{x \to 8} x)}^{3} - 1 \cdot 87}$

Step 14.3

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

$\frac{69 \cdot 8^{2} + 35 - 83 \cdot 8^{4}}{55 \cdot 8^{4} - 91 {(lim_{x \to 8} x)}^{3} - 1 \cdot 87}$

Step 14.4

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

$\frac{69 \cdot 8^{2} + 35 - 83 \cdot 8^{4}}{55 \cdot 8^{4} - 91 \cdot 8^{3} - 1 \cdot 87}$

Step 15

Simplify the answer.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

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

$\frac{69 \cdot 64 + 35 - 83 \cdot 8^{4}}{55 \cdot 8^{4} - 91 \cdot 8^{3} - 1 \cdot 87}$

Step 15.1.2

Multiply $69$ by $64$ .

$\frac{4416 + 35 - 83 \cdot 8^{4}}{55 \cdot 8^{4} - 91 \cdot 8^{3} - 1 \cdot 87}$

Step 15.1.3

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

$\frac{4416 + 35 - 83 \cdot 4096}{55 \cdot 8^{4} - 91 \cdot 8^{3} - 1 \cdot 87}$

Step 15.1.4

Multiply $- 83$ by $4096$ .

$\frac{4416 + 35 - 339968}{55 \cdot 8^{4} - 91 \cdot 8^{3} - 1 \cdot 87}$

Step 15.1.5

Add $4416$ and $35$ .

$\frac{4451 - 339968}{55 \cdot 8^{4} - 91 \cdot 8^{3} - 1 \cdot 87}$

Step 15.1.6

Subtract $339968$ from $4451$ .

$\frac{- 335517}{55 \cdot 8^{4} - 91 \cdot 8^{3} - 1 \cdot 87}$

Step 15.2

Simplify the denominator.

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Step 15.2.1

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

$\frac{- 335517}{55 \cdot 4096 - 91 \cdot 8^{3} - 1 \cdot 87}$

Step 15.2.2

Multiply $55$ by $4096$ .

$\frac{- 335517}{225280 - 91 \cdot 8^{3} - 1 \cdot 87}$

Step 15.2.3

Raise $8$ to the power of $3$ .

$\frac{- 335517}{225280 - 91 \cdot 512 - 1 \cdot 87}$

Step 15.2.4

Multiply $- 91$ by $512$ .

$\frac{- 335517}{225280 - 46592 - 1 \cdot 87}$

Step 15.2.5

Multiply $- 1$ by $87$ .

$\frac{- 335517}{225280 - 46592 - 87}$

Step 15.2.6

Subtract $46592$ from $225280$ .

$\frac{- 335517}{178688 - 87}$

Step 15.2.7

Subtract $87$ from $178688$ .

$\frac{- 335517}{178601}$

Step 15.3

Move the negative in front of the fraction.

$- \frac{335517}{178601}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- \frac{335517}{178601}$

Decimal Form:

$- 1.87858410 \dots$