Calculus Examples

$lim_{x \to 3} x e^{4 - x^{4}}$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $3$ .

$lim_{x \to 3} x \cdot lim_{x \to 3} e^{4 - x^{4}}$

Step 2

Move the limit into the exponent.

$lim_{x \to 3} x \cdot e^{lim_{x \to 3} 4 - x^{4}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$lim_{x \to 3} x \cdot e^{lim_{x \to 3} 4 - lim_{x \to 3} x^{4}}$

Step 4

Evaluate the limit of $4$ which is constant as $x$ approaches $3$ .

$lim_{x \to 3} x \cdot e^{4 - lim_{x \to 3} x^{4}}$

Step 5

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

$lim_{x \to 3} x \cdot e^{4 - {(lim_{x \to 3} x)}^{4}}$

Step 6

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$3 \cdot e^{4 - {(lim_{x \to 3} x)}^{4}}$

Step 6.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$3 \cdot e^{4 - 3^{4}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

$3 \cdot e^{4 - 1 \cdot 81}$

Step 7.1.2

Multiply $- 1$ by $81$ .

$3 \cdot e^{4 - 81}$

Step 7.2

Subtract $81$ from $4$ .

$3 \cdot e^{- 77}$

Step 7.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 \cdot \frac{1}{e^{77}}$

Step 7.4

Combine $3$ and $\frac{1}{e^{77}}$ .

$\frac{3}{e^{77}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{e^{77}}$

Decimal Form:

$1.08754227 \cdot 10^{- 33}$