Calculus Examples

$lim_{x \to 8} \frac{25 x^{2} + 3 x + 140}{5 x^{2 - 10}}$

Step 1

Move the term $\frac{1}{5}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} lim_{x \to 8} \frac{25 x^{2} + 3 x + 140}{x^{2 - 10}}$

Step 2

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

$\frac{1}{5} \cdot \frac{lim_{x \to 8} 25 x^{2} + 3 x + 140}{lim_{x \to 8} x^{2 - 10}}$

Step 3

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

$\frac{1}{5} \cdot \frac{lim_{x \to 8} 25 x^{2} + lim_{x \to 8} 3 x + lim_{x \to 8} 140}{lim_{x \to 8} x^{2 - 10}}$

Step 4

Move the term $25$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} \cdot \frac{25 lim_{x \to 8} x^{2} + lim_{x \to 8} 3 x + lim_{x \to 8} 140}{lim_{x \to 8} x^{2 - 10}}$

Step 5

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

$\frac{1}{5} \cdot \frac{25 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 3 x + lim_{x \to 8} 140}{lim_{x \to 8} x^{2 - 10}}$

Step 6

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

$\frac{1}{5} \cdot \frac{25 {(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x + lim_{x \to 8} 140}{lim_{x \to 8} x^{2 - 10}}$

Step 7

Evaluate the limit of $140$ which is constant as $x$ approaches $8$ .

$\frac{1}{5} \cdot \frac{25 {(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x + 140}{lim_{x \to 8} x^{2 - 10}}$

Step 8

Move the exponent $2 - 10$ from $x^{2 - 10}$ outside the limit using the Limits Power Rule.

$\frac{1}{5} \cdot \frac{25 {(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x + 140}{{(lim_{x \to 8} x)}^{2 - 10}}$

Step 9

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

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Step 9.1

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

$\frac{1}{5} \cdot \frac{25 \cdot 8^{2} + 3 lim_{x \to 8} x + 140}{{(lim_{x \to 8} x)}^{2 - 10}}$

Step 9.2

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

$\frac{1}{5} \cdot \frac{25 \cdot 8^{2} + 3 \cdot 8 + 140}{{(lim_{x \to 8} x)}^{2 - 10}}$

Step 9.3

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

$\frac{1}{5} \cdot \frac{25 \cdot 8^{2} + 3 \cdot 8 + 140}{8^{2 - 10}}$

Step 10

Simplify the answer.

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Step 10.1

Move $8^{2 - 10}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{1}{5} ((25 \cdot 8^{2} + 3 \cdot 8 + 140) \cdot 8^{- (2 - 10)})$

Step 10.2

Simplify each term.

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Step 10.2.1

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

$\frac{1}{5} ((25 \cdot 64 + 3 \cdot 8 + 140) \cdot 8^{- (2 - 10)})$

Step 10.2.2

Multiply $25$ by $64$ .

$\frac{1}{5} ((1600 + 3 \cdot 8 + 140) \cdot 8^{- (2 - 10)})$

Step 10.2.3

Multiply $3$ by $8$ .

$\frac{1}{5} ((1600 + 24 + 140) \cdot 8^{- (2 - 10)})$

Step 10.3

Add $1600$ and $24$ .

$\frac{1}{5} ((1624 + 140) \cdot 8^{- (2 - 10)})$

Step 10.4

Add $1624$ and $140$ .

$\frac{1}{5} (1764 \cdot 8^{- (2 - 10)})$

Step 10.5

Subtract $10$ from $2$ .

$\frac{1}{5} (1764 \cdot 8^{- - 8})$

Step 10.6

Multiply $- 1$ by $- 8$ .

$\frac{1}{5} (1764 \cdot 8^{8})$

Step 10.7

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

$\frac{1}{5} (1764 \cdot 16777216)$

Step 10.8

Multiply $1764$ by $16777216$ .

$\frac{1}{5} \cdot 29595009024$

Step 10.9

Combine $\frac{1}{5}$ and $29595009024$ .

$\frac{29595009024}{5}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{29595009024}{5}$

Decimal Form:

$5919001804.8$