Calculus Examples

$lim_{x \to 8} \frac{\sqrt{4 x^{4} + 3}}{5 x^{2} + 2}$

Step 1

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

$\frac{lim_{x \to 8} \sqrt{4 x^{4} + 3}}{lim_{x \to 8} 5 x^{2} + 2}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 4 x^{4} + 3}}{lim_{x \to 8} 5 x^{2} + 2}$

Step 3

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

$\frac{\sqrt{lim_{x \to 8} 4 x^{4} + lim_{x \to 8} 3}}{lim_{x \to 8} 5 x^{2} + 2}$

Step 4

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

$\frac{\sqrt{4 lim_{x \to 8} x^{4} + lim_{x \to 8} 3}}{lim_{x \to 8} 5 x^{2} + 2}$

Step 5

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

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{4} + lim_{x \to 8} 3}}{lim_{x \to 8} 5 x^{2} + 2}$

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{4} + 3}}{lim_{x \to 8} 5 x^{2} + 2}$

Step 7

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

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{4} + 3}}{lim_{x \to 8} 5 x^{2} + lim_{x \to 8} 2}$

Step 8

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{4} + 3}}{5 lim_{x \to 8} x^{2} + lim_{x \to 8} 2}$

Step 9

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

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{4} + 3}}{5 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 2}$

Step 10

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{4} + 3}}{5 {(lim_{x \to 8} x)}^{2} + 2}$

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 $x$ .

$\frac{\sqrt{4 \cdot 8^{4} + 3}}{5 {(lim_{x \to 8} x)}^{2} + 2}$

Step 11.2

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

$\frac{\sqrt{4 \cdot 8^{4} + 3}}{5 \cdot 8^{2} + 2}$

Step 12

Simplify the answer.

Tap for more steps...

Step 12.1

Simplify the numerator.

Tap for more steps...

Step 12.1.1

Raise $8$ to the power of $4$ .

$\frac{\sqrt{4 \cdot 4096 + 3}}{5 \cdot 8^{2} + 2}$

Step 12.1.2

Multiply $4$ by $4096$ .

$\frac{\sqrt{16384 + 3}}{5 \cdot 8^{2} + 2}$

Step 12.1.3

Add $16384$ and $3$ .

$\frac{\sqrt{16387}}{5 \cdot 8^{2} + 2}$

Step 12.2

Simplify the denominator.

Tap for more steps...

Step 12.2.1

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

$\frac{\sqrt{16387}}{5 \cdot 64 + 2}$

Step 12.2.2

Multiply $5$ by $64$ .

$\frac{\sqrt{16387}}{320 + 2}$

Step 12.2.3

Add $320$ and $2$ .

$\frac{\sqrt{16387}}{322}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{16387}}{322}$

Decimal Form:

$0.39755191 \dots$