Calculus Examples

$lim_{x \to 8} \frac{6 x^{7} + 4 x^{6} + 5}{4 x^{8}}$

Step 1

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} lim_{x \to 8} \frac{6 x^{7} + 4 x^{6} + 5}{x^{8}}$

Step 2

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

$\frac{1}{4} \cdot \frac{lim_{x \to 8} 6 x^{7} + 4 x^{6} + 5}{lim_{x \to 8} x^{8}}$

Step 3

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

$\frac{1}{4} \cdot \frac{lim_{x \to 8} 6 x^{7} + lim_{x \to 8} 4 x^{6} + lim_{x \to 8} 5}{lim_{x \to 8} x^{8}}$

Step 4

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

$\frac{1}{4} \cdot \frac{6 lim_{x \to 8} x^{7} + lim_{x \to 8} 4 x^{6} + lim_{x \to 8} 5}{lim_{x \to 8} x^{8}}$

Step 5

Move the exponent $7$ from $x^{7}$ outside the limit using the Limits Power Rule.

$\frac{1}{4} \cdot \frac{6 {(lim_{x \to 8} x)}^{7} + lim_{x \to 8} 4 x^{6} + lim_{x \to 8} 5}{lim_{x \to 8} x^{8}}$

Step 6

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} \cdot \frac{6 {(lim_{x \to 8} x)}^{7} + 4 lim_{x \to 8} x^{6} + lim_{x \to 8} 5}{lim_{x \to 8} x^{8}}$

Step 7

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

$\frac{1}{4} \cdot \frac{6 {(lim_{x \to 8} x)}^{7} + 4 {(lim_{x \to 8} x)}^{6} + lim_{x \to 8} 5}{lim_{x \to 8} x^{8}}$

Step 8

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$\frac{1}{4} \cdot \frac{6 {(lim_{x \to 8} x)}^{7} + 4 {(lim_{x \to 8} x)}^{6} + 5}{lim_{x \to 8} x^{8}}$

Step 9

Move the exponent $8$ from $x^{8}$ outside the limit using the Limits Power Rule.

$\frac{1}{4} \cdot \frac{6 {(lim_{x \to 8} x)}^{7} + 4 {(lim_{x \to 8} x)}^{6} + 5}{{(lim_{x \to 8} x)}^{8}}$

Step 10

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

Tap for more steps...

Step 10.1

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

$\frac{1}{4} \cdot \frac{6 \cdot 8^{7} + 4 {(lim_{x \to 8} x)}^{6} + 5}{{(lim_{x \to 8} x)}^{8}}$

Step 10.2

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

$\frac{1}{4} \cdot \frac{6 \cdot 8^{7} + 4 \cdot 8^{6} + 5}{{(lim_{x \to 8} x)}^{8}}$

Step 10.3

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

$\frac{1}{4} \cdot \frac{6 \cdot 8^{7} + 4 \cdot 8^{6} + 5}{8^{8}}$

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

Simplify the numerator.

Tap for more steps...

Step 11.1.1

Raise $8$ to the power of $7$ .

$\frac{1}{4} \cdot \frac{6 \cdot 2097152 + 4 \cdot 8^{6} + 5}{8^{8}}$

Step 11.1.2

Multiply $6$ by $2097152$ .

$\frac{1}{4} \cdot \frac{12582912 + 4 \cdot 8^{6} + 5}{8^{8}}$

Step 11.1.3

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

$\frac{1}{4} \cdot \frac{12582912 + 4 \cdot 262144 + 5}{8^{8}}$

Step 11.1.4

Multiply $4$ by $262144$ .

$\frac{1}{4} \cdot \frac{12582912 + 1048576 + 5}{8^{8}}$

Step 11.1.5

Add $12582912$ and $1048576$ .

$\frac{1}{4} \cdot \frac{13631488 + 5}{8^{8}}$

Step 11.1.6

Add $13631488$ and $5$ .

$\frac{1}{4} \cdot \frac{13631493}{8^{8}}$

Step 11.2

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

$\frac{1}{4} \cdot \frac{13631493}{16777216}$

Step 11.3

Multiply $\frac{1}{4} \cdot \frac{13631493}{16777216}$ .

Tap for more steps...

Step 11.3.1

Multiply $\frac{1}{4}$ by $\frac{13631493}{16777216}$ .

$\frac{13631493}{4 \cdot 16777216}$

Step 11.3.2

Multiply $4$ by $16777216$ .

$\frac{13631493}{67108864}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{13631493}{67108864}$

Decimal Form:

$0.20312507 \dots$