Calculus Examples

$lim_{x \to 8} 7 + \frac{11 x}{x^{2} - 2 x + 15}$

Step 1

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

$lim_{x \to 8} 7 + lim_{x \to 8} \frac{11 x}{x^{2} - 2 x + 15}$

Step 2

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$7 + lim_{x \to 8} \frac{11 x}{x^{2} - 2 x + 15}$

Step 3

Move the term $11$ outside of the limit because it is constant with respect to $x$ .

$7 + 11 lim_{x \to 8} \frac{x}{x^{2} - 2 x + 15}$

Step 4

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

$7 + 11 \frac{lim_{x \to 8} x}{lim_{x \to 8} x^{2} - 2 x + 15}$

Step 5

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

$7 + 11 \frac{lim_{x \to 8} x}{lim_{x \to 8} x^{2} - lim_{x \to 8} 2 x + lim_{x \to 8} 15}$

Step 6

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

$7 + 11 \frac{lim_{x \to 8} x}{{(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 2 x + lim_{x \to 8} 15}$

Step 7

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

$7 + 11 \frac{lim_{x \to 8} x}{{(lim_{x \to 8} x)}^{2} - 2 lim_{x \to 8} x + lim_{x \to 8} 15}$

Step 8

Evaluate the limit of $15$ which is constant as $x$ approaches $8$ .

$7 + 11 \frac{lim_{x \to 8} x}{{(lim_{x \to 8} x)}^{2} - 2 lim_{x \to 8} x + 15}$

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 $x$ .

$7 + 11 \frac{8}{{(lim_{x \to 8} x)}^{2} - 2 lim_{x \to 8} x + 15}$

Step 9.2

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

$7 + 11 \frac{8}{8^{2} - 2 lim_{x \to 8} x + 15}$

Step 9.3

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

$7 + 11 \frac{8}{8^{2} - 2 \cdot 8 + 15}$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

Simplify the denominator.

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Step 10.1.1.1

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

$7 + 11 \frac{8}{64 - 2 \cdot 8 + 15}$

Step 10.1.1.2

Multiply $- 2$ by $8$ .

$7 + 11 \frac{8}{64 - 16 + 15}$

Step 10.1.1.3

Subtract $16$ from $64$ .

$7 + 11 \frac{8}{48 + 15}$

Step 10.1.1.4

Add $48$ and $15$ .

$7 + 11 (\frac{8}{63})$

Step 10.1.2

Multiply $11 (\frac{8}{63})$ .

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Step 10.1.2.1

Combine $11$ and $\frac{8}{63}$ .

$7 + \frac{11 \cdot 8}{63}$

Step 10.1.2.2

Multiply $11$ by $8$ .

$7 + \frac{88}{63}$

Step 10.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{63}{63}$ .

$7 \cdot \frac{63}{63} + \frac{88}{63}$

Step 10.3

Combine $7$ and $\frac{63}{63}$ .

$\frac{7 \cdot 63}{63} + \frac{88}{63}$

Step 10.4

Combine the numerators over the common denominator.

$\frac{7 \cdot 63 + 88}{63}$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Multiply $7$ by $63$ .

$\frac{441 + 88}{63}$

Step 10.5.2

Add $441$ and $88$ .

$\frac{529}{63}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{529}{63}$

Decimal Form:

$8.39682539 \dots$