Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite $t^{2 {(t^{2} - 3 t + 4)}^{\frac{3}{2}}}$ as $e^{\ln (t^{2 {(t^{2} - 3 t + 4)}^{\frac{3}{2}}})}$ .

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

Step 1.2

Expand $\ln (t^{2 {(t^{2} - 3 t + 4)}^{\frac{3}{2}}})$ by moving $2 {(t^{2} - 3 t + 4)}^{\frac{3}{2}}$ outside the logarithm.

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

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

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

Step 2.3

Split the limit using the Product of Limits Rule on the limit as $t$ approaches $4$ .

$e^{2 lim_{t \to 4} {(t^{2} - 3 t + 4)}^{\frac{3}{2}} \cdot lim_{t \to 4} \ln (t)}$

Step 2.4

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

$e^{2 {(lim_{t \to 4} t^{2} - 3 t + 4)}^{\frac{3}{2}} \cdot lim_{t \to 4} \ln (t)}$

Step 2.5

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

$e^{2 {(lim_{t \to 4} t^{2} - lim_{t \to 4} 3 t + lim_{t \to 4} 4)}^{\frac{3}{2}} \cdot lim_{t \to 4} \ln (t)}$

Step 2.6

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

$e^{2 {({(lim_{t \to 4} t)}^{2} - lim_{t \to 4} 3 t + lim_{t \to 4} 4)}^{\frac{3}{2}} \cdot lim_{t \to 4} \ln (t)}$

Step 2.7

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

$e^{2 {({(lim_{t \to 4} t)}^{2} - 3 lim_{t \to 4} t + lim_{t \to 4} 4)}^{\frac{3}{2}} \cdot lim_{t \to 4} \ln (t)}$

Step 2.8

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

$e^{2 {({(lim_{t \to 4} t)}^{2} - 3 lim_{t \to 4} t + 4)}^{\frac{3}{2}} \cdot lim_{t \to 4} \ln (t)}$

Step 2.9

Move the limit inside the logarithm.

$e^{2 {({(lim_{t \to 4} t)}^{2} - 3 lim_{t \to 4} t + 4)}^{\frac{3}{2}} \cdot \ln (lim_{t \to 4} t)}$

Step 3

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

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Step 3.1

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

$e^{2 {(4^{2} - 3 lim_{t \to 4} t + 4)}^{\frac{3}{2}} \cdot \ln (lim_{t \to 4} t)}$

Step 3.2

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

$e^{2 {(4^{2} - 3 \cdot 4 + 4)}^{\frac{3}{2}} \cdot \ln (lim_{t \to 4} t)}$

Step 3.3

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $4$ to the power of $2$ .

$e^{2 {(16 - 3 \cdot 4 + 4)}^{\frac{3}{2}} \cdot \ln (4)}$

Step 4.1.2

Multiply $- 3$ by $4$ .

$e^{2 {(16 - 12 + 4)}^{\frac{3}{2}} \cdot \ln (4)}$

Step 4.2

Subtract $12$ from $16$ .

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

Step 4.3

Add $4$ and $4$ .

$e^{2 \cdot 8^{\frac{3}{2}} \cdot \ln (4)}$

Step 4.4

Multiply $2 \cdot 8^{\frac{3}{2}}$ .

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Step 4.4.1

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

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

Step 4.4.2

Multiply the exponents in ${(2^{3})}^{\frac{3}{2}}$ .

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Step 4.4.2.1

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

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

Step 4.4.2.2

Multiply $3 (\frac{3}{2})$ .

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Step 4.4.2.2.1

Combine $3$ and $\frac{3}{2}$ .

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

Step 4.4.2.2.2

Multiply $3$ by $3$ .

$e^{2 \cdot 2^{\frac{9}{2}} \cdot \ln (4)}$

Step 4.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{2^{1 + \frac{9}{2}} \cdot \ln (4)}$

Step 4.4.4

Write $1$ as a fraction with a common denominator.

$e^{2^{\frac{2}{2} + \frac{9}{2}} \cdot \ln (4)}$

Step 4.4.5

Combine the numerators over the common denominator.

$e^{2^{\frac{2 + 9}{2}} \cdot \ln (4)}$

Step 4.4.6

Add $2$ and $9$ .

$e^{2^{\frac{11}{2}} \cdot \ln (4)}$

$e^{2^{\frac{11}{2}} \ln (4)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$e^{2^{\frac{11}{2}} \ln (4)}$

Decimal Form:

$1.7624831 \cdot 10^{27}$