Calculus Examples

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

Step 1

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

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

Step 2

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

$- \frac{1}{3} \cdot 8^{3} + 8^{2}$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$- \frac{1}{3} \cdot 512 + 8^{2}$

Step 6.1.2

Multiply $- \frac{1}{3} \cdot 512$ .

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Step 6.1.2.1

Multiply $512$ by $- 1$ .

$- 512 (\frac{1}{3}) + 8^{2}$

Step 6.1.2.2

Combine $- 512$ and $\frac{1}{3}$ .

$\frac{- 512}{3} + 8^{2}$

Step 6.1.3

Move the negative in front of the fraction.

$- \frac{512}{3} + 8^{2}$

Step 6.1.4

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

$- \frac{512}{3} + 64$

Step 6.2

To write $64$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{512}{3} + 64 \cdot \frac{3}{3}$

Step 6.3

Combine $64$ and $\frac{3}{3}$ .

$- \frac{512}{3} + \frac{64 \cdot 3}{3}$

Step 6.4

Combine the numerators over the common denominator.

$\frac{- 512 + 64 \cdot 3}{3}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $64$ by $3$ .

$\frac{- 512 + 192}{3}$

Step 6.5.2

Add $- 512$ and $192$ .

$\frac{- 320}{3}$

Step 6.6

Move the negative in front of the fraction.

$- \frac{320}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{320}{3}$

Decimal Form:

$- 106.\overline{6}$