Calculus Examples

$lim_{z \to 4} z^{\frac{3}{2}} - 3 z^{\frac{1}{2}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $z$ approaches $4$ .

$lim_{z \to 4} z^{\frac{3}{2}} - lim_{z \to 4} 3 z^{\frac{1}{2}}$

Step 2

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

${(lim_{z \to 4} z)}^{\frac{3}{2}} - lim_{z \to 4} 3 z^{\frac{1}{2}}$

Step 3

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

${(lim_{z \to 4} z)}^{\frac{3}{2}} - 3 lim_{z \to 4} z^{\frac{1}{2}}$

Step 4

Move the exponent $\frac{1}{2}$ from $z^{\frac{1}{2}}$ outside the limit using the Limits Power Rule.

${(lim_{z \to 4} z)}^{\frac{3}{2}} - 3 {(lim_{z \to 4} z)}^{\frac{1}{2}}$

Step 5

Evaluate the limits by plugging in $4$ for all occurrences of $z$ .

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Step 5.1

Evaluate the limit of $z$ by plugging in $4$ for $z$ .

$4^{\frac{3}{2}} - 3 {(lim_{z \to 4} z)}^{\frac{1}{2}}$

Step 5.2

Evaluate the limit of $z$ by plugging in $4$ for $z$ .

$4^{\frac{3}{2}} - 3 \cdot 4^{\frac{1}{2}}$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

Rewrite $4$ as $2^{2}$ .

${(2^{2})}^{\frac{3}{2}} - 3 \cdot 4^{\frac{1}{2}}$

Step 6.1.2

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

$2^{2 (\frac{3}{2})} - 3 \cdot 4^{\frac{1}{2}}$

Step 6.1.3

Cancel the common factor of $2$ .

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Step 6.1.3.1

Cancel the common factor.

$2^{2 (\frac{3}{2})} - 3 \cdot 4^{\frac{1}{2}}$

Step 6.1.3.2

Rewrite the expression.

$2^{3} - 3 \cdot 4^{\frac{1}{2}}$

Step 6.1.4

Raise $2$ to the power of $3$ .

$8 - 3 \cdot 4^{\frac{1}{2}}$

Step 6.1.5

Rewrite $4$ as $2^{2}$ .

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

Step 6.1.6

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

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

Step 6.1.7

Cancel the common factor of $2$ .

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Step 6.1.7.1

Cancel the common factor.

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

Step 6.1.7.2

Rewrite the expression.

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

Step 6.1.8

Evaluate the exponent.

$8 - 3 \cdot 2$

Step 6.1.9

Multiply $- 3$ by $2$ .

$8 - 6$

Step 6.2

Subtract $6$ from $8$ .

$2$