Calculus Examples

$lim_{x \to 8} \frac{x + 3}{x^{2} + 7}$

Step 1

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

$\frac{lim_{x \to 8} x + 3}{lim_{x \to 8} x^{2} + 7}$

Step 2

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

$\frac{lim_{x \to 8} x + lim_{x \to 8} 3}{lim_{x \to 8} x^{2} + 7}$

Step 3

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x + 3}{lim_{x \to 8} x^{2} + 7}$

Step 4

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

$\frac{lim_{x \to 8} x + 3}{lim_{x \to 8} x^{2} + lim_{x \to 8} 7}$

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

$\frac{8 + 3}{8^{2} + 7}$

Step 8

Simplify the answer.

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Step 8.1

Add $8$ and $3$ .

$\frac{11}{8^{2} + 7}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

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

$\frac{11}{64 + 7}$

Step 8.2.2

Add $64$ and $7$ .

$\frac{11}{71}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{11}{71}$

Decimal Form:

$0.15492957 \dots$