Calculus Examples

$lim_{x \to 8} 3 x^{\frac{2}{3}} - 16 x^{- 1}$

Step 1

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

$lim_{x \to 8} 3 x^{\frac{2}{3}} - lim_{x \to 8} 16 x^{- 1}$

Step 2

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

$3 lim_{x \to 8} x^{\frac{2}{3}} - lim_{x \to 8} 16 x^{- 1}$

Step 3

Move the exponent $\frac{2}{3}$ from $x^{\frac{2}{3}}$ outside the limit using the Limits Power Rule.

$3 {(lim_{x \to 8} x)}^{\frac{2}{3}} - lim_{x \to 8} 16 x^{- 1}$

Step 4

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$3 {(lim_{x \to 8} x)}^{\frac{2}{3}} - 16 lim_{x \to 8} x^{- 1}$

Step 5

Move the exponent $- 1$ from $x^{- 1}$ outside the limit using the Limits Power Rule.

$3 {(lim_{x \to 8} x)}^{\frac{2}{3}} - 16 {(lim_{x \to 8} x)}^{- 1}$

Step 6

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

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Step 6.1

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

$3 \cdot 8^{\frac{2}{3}} - 16 {(lim_{x \to 8} x)}^{- 1}$

Step 6.2

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

$3 \cdot 8^{\frac{2}{3}} - 16 \cdot 8^{- 1}$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Rewrite $8$ as $2^{3}$ .

$3 \cdot {(2^{3})}^{\frac{2}{3}} - 16 \cdot 8^{- 1}$

Step 7.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 \cdot 2^{3 (\frac{2}{3})} - 16 \cdot 8^{- 1}$

Step 7.1.3

Cancel the common factor of $3$ .

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Step 7.1.3.1

Cancel the common factor.

$3 \cdot 2^{3 (\frac{2}{3})} - 16 \cdot 8^{- 1}$

Step 7.1.3.2

Rewrite the expression.

$3 \cdot 2^{2} - 16 \cdot 8^{- 1}$

Step 7.1.4

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

$3 \cdot 4 - 16 \cdot 8^{- 1}$

Step 7.1.5

Multiply $3$ by $4$ .

$12 - 16 \cdot 8^{- 1}$

Step 7.1.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$12 - 16 \cdot \frac{1}{8}$

Step 7.1.7

Cancel the common factor of $8$ .

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Step 7.1.7.1

Factor $8$ out of $- 16$ .

$12 + 8 (- 2) \cdot \frac{1}{8}$

Step 7.1.7.2

Cancel the common factor.

$12 + 8 \cdot - 2 \cdot \frac{1}{8}$

Step 7.1.7.3

Rewrite the expression.

$12 - 2$

Step 7.2

Subtract $2$ from $12$ .

$10$