Calculus Examples

$lim_{x \to 8} \frac{2 x^{2}}{e^{6 x}}$

Step 1

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

$2 lim_{x \to 8} \frac{x^{2}}{e^{6 x}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$2 \frac{lim_{x \to 8} x^{2}}{lim_{x \to 8} e^{6 x}}$

Step 3

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

$2 \frac{{(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} e^{6 x}}$

Step 4

Move the limit into the exponent.

$2 \frac{{(lim_{x \to 8} x)}^{2}}{e^{lim_{x \to 8} 6 x}}$

Step 5

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{{(lim_{x \to 8} x)}^{2}}{e^{6 lim_{x \to 8} x}}$

Step 6

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$2 \frac{8^{2}}{e^{6 lim_{x \to 8} x}}$

Step 6.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$2 \frac{8^{2}}{e^{6 \cdot 8}}$

Step 7

Simplify the answer.

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Step 7.1

Raise $8$ to the power of $2$ .

$2 \frac{64}{e^{6 \cdot 8}}$

Step 7.2

Multiply $6$ by $8$ .

$2 \frac{64}{e^{48}}$

Step 7.3

Multiply $2 \frac{64}{e^{48}}$ .

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Step 7.3.1

Combine $2$ and $\frac{64}{e^{48}}$ .

$\frac{2 \cdot 64}{e^{48}}$

Step 7.3.2

Multiply $2$ by $64$ .

$\frac{128}{e^{48}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{128}{e^{48}}$

Decimal Form:

$1.82421002 \cdot 10^{- 19}$