Calculus Examples

$lim_{x \to 8} \frac{6 x - 21 x^{3} + 11}{\sqrt{x} + 3 x^{3}}$

Step 1

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

$\frac{lim_{x \to 8} 6 x - 21 x^{3} + 11}{lim_{x \to 8} \sqrt{x} + 3 x^{3}}$

Step 2

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

$\frac{lim_{x \to 8} 6 x - lim_{x \to 8} 21 x^{3} + lim_{x \to 8} 11}{lim_{x \to 8} \sqrt{x} + 3 x^{3}}$

Step 3

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

$\frac{6 lim_{x \to 8} x - lim_{x \to 8} 21 x^{3} + lim_{x \to 8} 11}{lim_{x \to 8} \sqrt{x} + 3 x^{3}}$

Step 4

Move the term $21$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 lim_{x \to 8} x - 21 lim_{x \to 8} x^{3} + lim_{x \to 8} 11}{lim_{x \to 8} \sqrt{x} + 3 x^{3}}$

Step 5

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{6 lim_{x \to 8} x - 21 {(lim_{x \to 8} x)}^{3} + lim_{x \to 8} 11}{lim_{x \to 8} \sqrt{x} + 3 x^{3}}$

Step 6

Evaluate the limit of $11$ which is constant as $x$ approaches $8$ .

$\frac{6 lim_{x \to 8} x - 21 {(lim_{x \to 8} x)}^{3} + 11}{lim_{x \to 8} \sqrt{x} + 3 x^{3}}$

Step 7

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

$\frac{6 lim_{x \to 8} x - 21 {(lim_{x \to 8} x)}^{3} + 11}{lim_{x \to 8} \sqrt{x} + lim_{x \to 8} 3 x^{3}}$

Step 8

Move the limit under the radical sign.

$\frac{6 lim_{x \to 8} x - 21 {(lim_{x \to 8} x)}^{3} + 11}{\sqrt{lim_{x \to 8} x} + lim_{x \to 8} 3 x^{3}}$

Step 9

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 lim_{x \to 8} x - 21 {(lim_{x \to 8} x)}^{3} + 11}{\sqrt{lim_{x \to 8} x} + 3 lim_{x \to 8} x^{3}}$

Step 10

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{6 lim_{x \to 8} x - 21 {(lim_{x \to 8} x)}^{3} + 11}{\sqrt{lim_{x \to 8} x} + 3 {(lim_{x \to 8} x)}^{3}}$

Step 11

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

Tap for more steps...

Step 11.1

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

$\frac{6 \cdot 8 - 21 {(lim_{x \to 8} x)}^{3} + 11}{\sqrt{lim_{x \to 8} x} + 3 {(lim_{x \to 8} x)}^{3}}$

Step 11.2

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

$\frac{6 \cdot 8 - 21 \cdot 8^{3} + 11}{\sqrt{lim_{x \to 8} x} + 3 {(lim_{x \to 8} x)}^{3}}$

Step 11.3

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

$\frac{6 \cdot 8 - 21 \cdot 8^{3} + 11}{\sqrt{8} + 3 {(lim_{x \to 8} x)}^{3}}$

Step 11.4

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

$\frac{6 \cdot 8 - 21 \cdot 8^{3} + 11}{\sqrt{8} + 3 \cdot 8^{3}}$

Step 12

Simplify the answer.

Tap for more steps...

Step 12.1

Simplify the numerator.

Tap for more steps...

Step 12.1.1

Multiply $6$ by $8$ .

$\frac{48 - 21 \cdot 8^{3} + 11}{\sqrt{8} + 3 \cdot 8^{3}}$

Step 12.1.2

Raise $8$ to the power of $3$ .

$\frac{48 - 21 \cdot 512 + 11}{\sqrt{8} + 3 \cdot 8^{3}}$

Step 12.1.3

Multiply $- 21$ by $512$ .

$\frac{48 - 10752 + 11}{\sqrt{8} + 3 \cdot 8^{3}}$

Step 12.1.4

Subtract $10752$ from $48$ .

$\frac{- 10704 + 11}{\sqrt{8} + 3 \cdot 8^{3}}$

Step 12.1.5

Add $- 10704$ and $11$ .

$\frac{- 10693}{\sqrt{8} + 3 \cdot 8^{3}}$

Step 12.2

Simplify the denominator.

Tap for more steps...

Step 12.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 12.2.1.1

Factor $4$ out of $8$ .

$\frac{- 10693}{\sqrt{4 (2)} + 3 \cdot 8^{3}}$

Step 12.2.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{- 10693}{\sqrt{2^{2} \cdot 2} + 3 \cdot 8^{3}}$

Step 12.2.2

Pull terms out from under the radical.

$\frac{- 10693}{2 \sqrt{2} + 3 \cdot 8^{3}}$

Step 12.2.3

Raise $8$ to the power of $3$ .

$\frac{- 10693}{2 \sqrt{2} + 3 \cdot 512}$

Step 12.2.4

Multiply $3$ by $512$ .

$\frac{- 10693}{2 \sqrt{2} + 1536}$

Step 12.3

Move the negative in front of the fraction.

$- \frac{10693}{2 \sqrt{2} + 1536}$

Step 12.4

Multiply $\frac{10693}{2 \sqrt{2} + 1536}$ by $\frac{2 \sqrt{2} - 1536}{2 \sqrt{2} - 1536}$ .

$- (\frac{10693}{2 \sqrt{2} + 1536} \cdot \frac{2 \sqrt{2} - 1536}{2 \sqrt{2} - 1536})$

Step 12.5

Multiply $\frac{10693}{2 \sqrt{2} + 1536}$ by $\frac{2 \sqrt{2} - 1536}{2 \sqrt{2} - 1536}$ .

$- \frac{10693 (2 \sqrt{2} - 1536)}{(2 \sqrt{2} + 1536) (2 \sqrt{2} - 1536)}$

Step 12.6

Expand the denominator using the FOIL method.

$- \frac{10693 (2 \sqrt{2} - 1536)}{4 {\sqrt{2}}^{2} - 3072 \sqrt{2} + 3072 \sqrt{2} - 2359296}$

Step 12.7

Simplify.

$- \frac{10693 (2 \sqrt{2} - 1536)}{- 2359288}$

Step 12.8

Cancel the common factor of $2 \sqrt{2} - 1536$ and $- 2359288$ .

Tap for more steps...

Step 12.8.1

Factor $2$ out of $10693 (2 \sqrt{2} - 1536)$ .

$- \frac{2 (10693 (\sqrt{2} - 768))}{- 2359288}$

Step 12.8.2

Cancel the common factors.

Tap for more steps...

Step 12.8.2.1

Factor $2$ out of $- 2359288$ .

$- \frac{2 (10693 (\sqrt{2} - 768))}{2 (- 1179644)}$

Step 12.8.2.2

Cancel the common factor.

$- \frac{2 (10693 (\sqrt{2} - 768))}{2 \cdot - 1179644}$

Step 12.8.2.3

Rewrite the expression.

$- \frac{10693 (\sqrt{2} - 768)}{- 1179644}$

Step 12.9

Move the negative in front of the fraction.

$- - \frac{10693 (\sqrt{2} - 768)}{1179644}$

Step 12.10

Multiply $- - \frac{10693 (\sqrt{2} - 768)}{1179644}$ .

Tap for more steps...

Step 12.10.1

Multiply $- 1$ by $- 1$ .

$1 \frac{10693 (\sqrt{2} - 768)}{1179644}$

Step 12.10.2

Multiply $\frac{10693 (\sqrt{2} - 768)}{1179644}$ by $1$ .

$\frac{10693 (\sqrt{2} - 768)}{1179644}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{10693 (\sqrt{2} - 768)}{1179644}$

Decimal Form:

$- 6.94879286 \dots$