Calculus Examples

$lim_{t \to \frac{3}{2}} {(4 t^{2} + 8 t - 5)}^{\frac{3}{2}}$

Step 1

Move the exponent $\frac{3}{2}$ from ${(4 t^{2} + 8 t - 5)}^{\frac{3}{2}}$ outside the limit using the Limits Power Rule.

${(lim_{t \to \frac{3}{2}} 4 t^{2} + 8 t - 5)}^{\frac{3}{2}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $\frac{3}{2}$ .

${(lim_{t \to \frac{3}{2}} 4 t^{2} + lim_{t \to \frac{3}{2}} 8 t - lim_{t \to \frac{3}{2}} 5)}^{\frac{3}{2}}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $t$ .

${(4 lim_{t \to \frac{3}{2}} t^{2} + lim_{t \to \frac{3}{2}} 8 t - lim_{t \to \frac{3}{2}} 5)}^{\frac{3}{2}}$

Step 4

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

${(4 {(lim_{t \to \frac{3}{2}} t)}^{2} + lim_{t \to \frac{3}{2}} 8 t - lim_{t \to \frac{3}{2}} 5)}^{\frac{3}{2}}$

Step 5

Move the term $8$ outside of the limit because it is constant with respect to $t$ .

${(4 {(lim_{t \to \frac{3}{2}} t)}^{2} + 8 lim_{t \to \frac{3}{2}} t - lim_{t \to \frac{3}{2}} 5)}^{\frac{3}{2}}$

Step 6

Evaluate the limit of $5$ which is constant as $t$ approaches $\frac{3}{2}$ .

${(4 {(lim_{t \to \frac{3}{2}} t)}^{2} + 8 lim_{t \to \frac{3}{2}} t - 1 \cdot 5)}^{\frac{3}{2}}$

Step 7

Evaluate the limits by plugging in $\frac{3}{2}$ for all occurrences of $t$ .

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Step 7.1

Evaluate the limit of $t$ by plugging in $\frac{3}{2}$ for $t$ .

${(4 {(\frac{3}{2})}^{2} + 8 lim_{t \to \frac{3}{2}} t - 1 \cdot 5)}^{\frac{3}{2}}$

Step 7.2

Evaluate the limit of $t$ by plugging in $\frac{3}{2}$ for $t$ .

${(4 {(\frac{3}{2})}^{2} + 8 (\frac{3}{2}) - 1 \cdot 5)}^{\frac{3}{2}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

Apply the product rule to $\frac{3}{2}$ .

${(4 \frac{3^{2}}{2^{2}} + 8 (\frac{3}{2}) - 1 \cdot 5)}^{\frac{3}{2}}$

Step 8.1.2

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

${(4 \frac{9}{2^{2}} + 8 (\frac{3}{2}) - 1 \cdot 5)}^{\frac{3}{2}}$

Step 8.1.3

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

${(4 (\frac{9}{4}) + 8 (\frac{3}{2}) - 1 \cdot 5)}^{\frac{3}{2}}$

Step 8.1.4

Cancel the common factor of $4$ .

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Step 8.1.4.1

Cancel the common factor.

${(4 (\frac{9}{4}) + 8 (\frac{3}{2}) - 1 \cdot 5)}^{\frac{3}{2}}$

Step 8.1.4.2

Rewrite the expression.

${(9 + 8 (\frac{3}{2}) - 1 \cdot 5)}^{\frac{3}{2}}$

Step 8.1.5

Cancel the common factor of $2$ .

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Step 8.1.5.1

Factor $2$ out of $8$ .

${(9 + 2 (4) \frac{3}{2} - 1 \cdot 5)}^{\frac{3}{2}}$

Step 8.1.5.2

Cancel the common factor.

${(9 + 2 \cdot 4 \frac{3}{2} - 1 \cdot 5)}^{\frac{3}{2}}$

Step 8.1.5.3

Rewrite the expression.

${(9 + 4 \cdot 3 - 1 \cdot 5)}^{\frac{3}{2}}$

Step 8.1.6

Multiply $4$ by $3$ .

${(9 + 12 - 1 \cdot 5)}^{\frac{3}{2}}$

Step 8.1.7

Multiply $- 1$ by $5$ .

${(9 + 12 - 5)}^{\frac{3}{2}}$

Step 8.2

Add $9$ and $12$ .

${(21 - 5)}^{\frac{3}{2}}$

Step 8.3

Subtract $5$ from $21$ .

$16^{\frac{3}{2}}$

Step 8.4

Rewrite $16$ as $4^{2}$ .

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

Step 8.5

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

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

Step 8.6

Cancel the common factor of $2$ .

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Step 8.6.1

Cancel the common factor.

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

Step 8.6.2

Rewrite the expression.

$4^{3}$

Step 8.7

Raise $4$ to the power of $3$ .

$64$