Calculus Examples

$lim_{x \to \infty} \frac{\sqrt{5 x^{2} + 2 x}}{x}$

Step 1

Factor $x$ out of $5 x^{2} + 2 x$ .

Tap for more steps...

Step 1.1

Factor $x$ out of $5 x^{2}$ .

$lim_{x \to \infty} \frac{\sqrt{x (5 x) + 2 x}}{x}$

Step 1.2

Factor $x$ out of $2 x$ .

$lim_{x \to \infty} \frac{\sqrt{x (5 x) + x \cdot 2}}{x}$

Step 1.3

Factor $x$ out of $x (5 x) + x \cdot 2$ .

$lim_{x \to \infty} \frac{\sqrt{x (5 x + 2)}}{x}$

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $\sqrt{x^{2}} = x$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{x (5 x + 2)}{x^{2}}}}{\frac{x}{x}}$

Step 3

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{x (5 x + 2)}{x^{2}}}}{1}$

Step 4

Cancel the common factors.

Tap for more steps...

Step 4.1

Factor $x$ out of $x^{2}$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{x (5 x + 2)}{x \cdot x}}}{1}$

Step 4.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{\frac{x (5 x + 2)}{x \cdot x}}}{1}$

Step 4.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\sqrt{\frac{5 x + 2}{x}}}{1}$

Step 5

Evaluate the limit.

Tap for more steps...

Step 5.1

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

$\frac{lim_{x \to \infty} \sqrt{\frac{5 x + 2}{x}}}{lim_{x \to \infty} 1}$

Step 5.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to \infty} \frac{5 x + 2}{x}}}{lim_{x \to \infty} 1}$

Step 6

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x$ .

$\frac{\sqrt{lim_{x \to \infty} \frac{\frac{5 x}{x} + \frac{2}{x}}{\frac{x}{x}}}}{lim_{x \to \infty} 1}$

Step 7

Evaluate the limit.

Tap for more steps...

Step 7.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 7.1.1

Cancel the common factor.

$\frac{\sqrt{lim_{x \to \infty} \frac{\frac{5 x}{x} + \frac{2}{x}}{\frac{x}{x}}}}{lim_{x \to \infty} 1}$

Step 7.1.2

Divide $5$ by $1$ .

$\frac{\sqrt{lim_{x \to \infty} \frac{5 + \frac{2}{x}}{\frac{x}{x}}}}{lim_{x \to \infty} 1}$

Step 7.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 7.2.1

Cancel the common factor.

$\frac{\sqrt{lim_{x \to \infty} \frac{5 + \frac{2}{x}}{\frac{x}{x}}}}{lim_{x \to \infty} 1}$

Step 7.2.2

Rewrite the expression.

$\frac{\sqrt{lim_{x \to \infty} \frac{5 + \frac{2}{x}}{1}}}{lim_{x \to \infty} 1}$

Step 7.3

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

$\frac{\sqrt{\frac{lim_{x \to \infty} 5 + \frac{2}{x}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1}$

Step 7.4

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

$\frac{\sqrt{\frac{lim_{x \to \infty} 5 + lim_{x \to \infty} \frac{2}{x}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1}$

Step 7.5

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

$\frac{\sqrt{\frac{5 + lim_{x \to \infty} \frac{2}{x}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1}$

Step 7.6

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

$\frac{\sqrt{\frac{5 + 2 lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1}$

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{\sqrt{\frac{5 + 2 \cdot 0}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1}$

Step 9

Evaluate the limit.

Tap for more steps...

Step 9.1

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{\frac{5 + 2 \cdot 0}{1}}}{lim_{x \to \infty} 1}$

Step 9.2

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{\frac{5 + 2 \cdot 0}{1}}}{1}$

Step 9.3

Simplify the answer.

Tap for more steps...

Step 9.3.1

Divide $5 + 2 \cdot 0$ by $1$ .

$\frac{\sqrt{5 + 2 \cdot 0}}{1}$

Step 9.3.2

Divide $\sqrt{5 + 2 \cdot 0}$ by $1$ .

$\sqrt{5 + 2 \cdot 0}$

Step 9.3.3

Multiply $2$ by $0$ .

$\sqrt{5 + 0}$

Step 9.3.4

Add $5$ and $0$ .

$\sqrt{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\sqrt{5}$

Decimal Form:

$2.23606797 \dots$