Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the limit under the radical sign.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Move the limit under the radical sign.

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

Step 9

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

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

Step 10

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

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

Step 11

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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Step 15.1