Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

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

Step 5

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

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

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 $x$ .

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

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

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

$(64 + 1) \cdot 8^{\frac{2}{3}}$

Step 7.2

Add $64$ and $1$ .

$65 \cdot 8^{\frac{2}{3}}$

Step 7.3

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

$65 \cdot {(2^{3})}^{\frac{2}{3}}$

Step 7.4

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

$65 \cdot 2^{3 (\frac{2}{3})}$

Step 7.5

Cancel the common factor of $3$ .

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Step 7.5.1

Cancel the common factor.

$65 \cdot 2^{3 (\frac{2}{3})}$

Step 7.5.2

Rewrite the expression.

$65 \cdot 2^{2}$

Step 7.6

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

$65 \cdot 4$

Step 7.7

Multiply $65$ by $4$ .

$260$