Calculus Examples

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

Step 1

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

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

Step 2

Move the limit into the exponent.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Evaluate the limits by plugging in $\frac{1}{\frac{}{}} \cdot 2$ for all occurrences of $x$ .

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Step 6.1

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

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

Step 6.2

Reduce the expression by cancelling the common factors.

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Step 6.2.1

Reduce the expression $\frac{}{}$ by cancelling the common factors.

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Rewrite the expression.

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

Step 6.2.2

Rewrite the expression.

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

Step 6.3

Cancel the common factor of $1$ .

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Step 6.3.1

Cancel the common factor.

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

Step 6.3.2

Rewrite the expression.

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

Step 6.4

Multiply $2$ by $1$ .

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Combine $16$ and $\frac{9}{9}$ .

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

Step 7.4

Combine the numerators over the common denominator.

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

Step 7.5

Simplify the numerator.

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Step 7.5.1

Multiply $16$ by $9$ .

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

Step 7.5.2

Add $144$ and $2$ .

$2 e^{\frac{146}{9}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 e^{\frac{146}{9}}$

Decimal Form:

$22194818.1467204$