Calculus Examples

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

Step 1

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

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

Step 2

Move the exponent $6$ from $x^{6}$ outside the limit using the Limits Power Rule.

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

Step 3

Move the limit into the exponent.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

$\frac{8^{6}}{6^{8}}$

Step 5

Simplify the answer.

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Step 5.1

Raise $8$ to the power of $6$ .

$\frac{262144}{6^{8}}$

Step 5.2

Raise $6$ to the power of $8$ .

$\frac{262144}{1679616}$

Step 5.3

Cancel the common factor of $262144$ and $1679616$ .

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Step 5.3.1

Factor $256$ out of $262144$ .

$\frac{256 (1024)}{1679616}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $256$ out of $1679616$ .

$\frac{256 \cdot 1024}{256 \cdot 6561}$

Step 5.3.2.2

Cancel the common factor.

$\frac{256 \cdot 1024}{256 \cdot 6561}$

Step 5.3.2.3

Rewrite the expression.

$\frac{1024}{6561}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1024}{6561}$

Decimal Form:

$0.15607376 \dots$