Calculus Examples

$\frac{lim_{x \to 8} x + x^{3} + x^{5}}{1 - x^{2} + x^{4}}$

Step 1

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

$\frac{lim_{x \to 8} x + lim_{x \to 8} x^{3} + lim_{x \to 8} x^{5}}{1 - x^{2} + x^{4}}$

Step 2

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

$\frac{lim_{x \to 8} x + {(lim_{x \to 8} x)}^{3} + lim_{x \to 8} x^{5}}{1 - x^{2} + x^{4}}$

Step 3

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{lim_{x \to 8} x + {(lim_{x \to 8} x)}^{3} + {(lim_{x \to 8} x)}^{5}}{1 - x^{2} + x^{4}}$

Step 4

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

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Step 4.1

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

$\frac{8 + {(lim_{x \to 8} x)}^{3} + {(lim_{x \to 8} x)}^{5}}{1 - x^{2} + x^{4}}$

Step 4.2

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

$\frac{8 + 8^{3} + {(lim_{x \to 8} x)}^{5}}{1 - x^{2} + x^{4}}$

Step 4.3

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

$\frac{8 + 8^{3} + 8^{5}}{1 - x^{2} + x^{4}}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{8 + 512 + 8^{5}}{1 - x^{2} + x^{4}}$

Step 5.2

Raise $8$ to the power of $5$ .

$\frac{8 + 512 + 32768}{1 - x^{2} + x^{4}}$

Step 5.3

Add $8$ and $512$ .

$\frac{520 + 32768}{1 - x^{2} + x^{4}}$

Step 5.4

Add $520$ and $32768$ .

$\frac{33288}{1 - x^{2} + x^{4}}$