Calculus Examples

$lim_{z \to 2} {(\frac{2 z + z^{2}}{z^{2} + 4})}^{3}$

Step 1

Move the exponent $3$ from ${(\frac{2 z + z^{2}}{z^{2} + 4})}^{3}$ outside the limit using the Limits Power Rule.

${(lim_{z \to 2} \frac{2 z + z^{2}}{z^{2} + 4})}^{3}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $z$ approaches $2$ .

${(\frac{lim_{z \to 2} 2 z + z^{2}}{lim_{z \to 2} z^{2} + 4})}^{3}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $z$ approaches $2$ .

${(\frac{lim_{z \to 2} 2 z + lim_{z \to 2} z^{2}}{lim_{z \to 2} z^{2} + 4})}^{3}$

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $z$ .

${(\frac{2 lim_{z \to 2} z + lim_{z \to 2} z^{2}}{lim_{z \to 2} z^{2} + 4})}^{3}$

Step 5

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

${(\frac{2 lim_{z \to 2} z + {(lim_{z \to 2} z)}^{2}}{lim_{z \to 2} z^{2} + 4})}^{3}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $z$ approaches $2$ .

${(\frac{2 lim_{z \to 2} z + {(lim_{z \to 2} z)}^{2}}{lim_{z \to 2} z^{2} + lim_{z \to 2} 4})}^{3}$

Step 7

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

${(\frac{2 lim_{z \to 2} z + {(lim_{z \to 2} z)}^{2}}{{(lim_{z \to 2} z)}^{2} + lim_{z \to 2} 4})}^{3}$

Step 8

Evaluate the limit of $4$ which is constant as $z$ approaches $2$ .

${(\frac{2 lim_{z \to 2} z + {(lim_{z \to 2} z)}^{2}}{{(lim_{z \to 2} z)}^{2} + 4})}^{3}$

Step 9

Evaluate the limits by plugging in $2$ for all occurrences of $z$ .

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Step 9.1

Evaluate the limit of $z$ by plugging in $2$ for $z$ .

${(\frac{2 \cdot 2 + {(lim_{z \to 2} z)}^{2}}{{(lim_{z \to 2} z)}^{2} + 4})}^{3}$

Step 9.2

Evaluate the limit of $z$ by plugging in $2$ for $z$ .

${(\frac{2 \cdot 2 + 2^{2}}{{(lim_{z \to 2} z)}^{2} + 4})}^{3}$

Step 9.3

Evaluate the limit of $z$ by plugging in $2$ for $z$ .

${(\frac{2 \cdot 2 + 2^{2}}{2^{2} + 4})}^{3}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Multiply $2$ by $2$ .

${(\frac{4 + 2^{2}}{2^{2} + 4})}^{3}$

Step 10.1.2

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

${(\frac{4 + 4}{2^{2} + 4})}^{3}$

Step 10.1.3

Add $4$ and $4$ .

${(\frac{8}{2^{2} + 4})}^{3}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

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

${(\frac{8}{4 + 4})}^{3}$

Step 10.2.2

Add $4$ and $4$ .

${(\frac{8}{8})}^{3}$

Step 10.3

Divide $8$ by $8$ .

$1^{3}$

Step 10.4

One to any power is one.

$1$