Calculus Examples

$lim_{x \to \frac{1}{2}} 2 e^{x^{4} + \frac{2}{9}}$

Step 1

Evaluate the limit.

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Step 1.1

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

$2 lim_{x \to \frac{1}{2}} e^{x^{4} + \frac{2}{9}}$

Step 1.2

Move the limit into the exponent.

$2 e^{lim_{x \to \frac{1}{2}} x^{4} + \frac{2}{9}}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$2 e^{lim_{x \to \frac{1}{2}} x^{4} + lim_{x \to \frac{1}{2}} \frac{2}{9}}$

Step 1.4

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$2 e^{{(lim_{x \to \frac{1}{2}} x)}^{4} + lim_{x \to \frac{1}{2}} \frac{2}{9}}$

Step 1.5

Evaluate the limit of $\frac{2}{9}$ which is constant as $x$ approaches $\frac{1}{2}$ .

$2 e^{{(lim_{x \to \frac{1}{2}} x)}^{4} + \frac{2}{9}}$

Step 2

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

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

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the product rule to $\frac{1}{2}$ .

$2 e^{\frac{1^{4}}{2^{4}} + \frac{2}{9}}$

Step 3.1.2

One to any power is one.

$2 e^{\frac{1}{2^{4}} + \frac{2}{9}}$

Step 3.1.3

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

$2 e^{\frac{1}{16} + \frac{2}{9}}$

Step 3.2

To write $\frac{1}{16}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$2 e^{\frac{1}{16} \cdot \frac{9}{9} + \frac{2}{9}}$

Step 3.3

To write $\frac{2}{9}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$2 e^{\frac{1}{16} \cdot \frac{9}{9} + \frac{2}{9} \cdot \frac{16}{16}}$

Step 3.4

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{1}{16}$ by $\frac{9}{9}$ .

$2 e^{\frac{9}{16 \cdot 9} + \frac{2}{9} \cdot \frac{16}{16}}$

Step 3.4.2

Multiply $16$ by $9$ .

$2 e^{\frac{9}{144} + \frac{2}{9} \cdot \frac{16}{16}}$

Step 3.4.3

Multiply $\frac{2}{9}$ by $\frac{16}{16}$ .

$2 e^{\frac{9}{144} + \frac{2 \cdot 16}{9 \cdot 16}}$

Step 3.4.4

Multiply $9$ by $16$ .

$2 e^{\frac{9}{144} + \frac{2 \cdot 16}{144}}$

Step 3.5

Combine the numerators over the common denominator.

$2 e^{\frac{9 + 2 \cdot 16}{144}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $2$ by $16$ .

$2 e^{\frac{9 + 32}{144}}$

Step 3.6.2

Add $9$ and $32$ .

$2 e^{\frac{41}{144}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$2 e^{\frac{41}{144}}$

Decimal Form:

$2.65878540 \dots$