Calculus Examples

$lim_{x \to \infty} \frac{\sqrt{6 x + 5 x^{2}}}{6 x - 1}$

Step 1

Factor $x$ out of $6 x + 5 x^{2}$ .

Tap for more steps...

Step 1.1

Factor $x$ out of $6 x$ .

$lim_{x \to \infty} \frac{\sqrt{x \cdot 6 + 5 x^{2}}}{6 x - 1}$

Step 1.2

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

$lim_{x \to \infty} \frac{\sqrt{x \cdot 6 + x (5 x)}}{6 x - 1}$

Step 1.3

Factor $x$ out of $x \cdot 6 + x (5 x)$ .

$lim_{x \to \infty} \frac{\sqrt{x (6 + 5 x)}}{6 x - 1}$

Step 2

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

$lim_{x \to \infty} \frac{\sqrt{\frac{x (6 + 5 x)}{x^{2}}}}{\frac{6 x}{x} - \frac{1}{x}}$

Step 3

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{x (6 + 5 x)}{x^{2}}}}{6 - \frac{1}{x}}$

Step 4

Cancel the common factors.

Tap for more steps...

Step 4.1

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

$lim_{x \to \infty} \frac{\sqrt{\frac{x (6 + 5 x)}{x \cdot x}}}{6 - \frac{1}{x}}$

Step 4.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{\frac{x (6 + 5 x)}{x \cdot x}}}{6 - \frac{1}{x}}$

Step 4.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\sqrt{\frac{6 + 5 x}{x}}}{6 - \frac{1}{x}}$

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 $\infty$ .

$\frac{lim_{x \to \infty} \sqrt{\frac{6 + 5 x}{x}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 5.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to \infty} \frac{6 + 5 x}{x}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 6

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

$\frac{\sqrt{lim_{x \to \infty} \frac{\frac{6}{x} + \frac{5 x}{x}}{\frac{x}{x}}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

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.

$\frac{\sqrt{lim_{x \to \infty} \frac{\frac{6}{x} + \frac{5 x}{x}}{\frac{x}{x}}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 7.1.2

Divide $5$ by $1$ .

$\frac{\sqrt{lim_{x \to \infty} \frac{\frac{6}{x} + 5}{\frac{x}{x}}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 7.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 7.2.1

Cancel the common factor.

$\frac{\sqrt{lim_{x \to \infty} \frac{\frac{6}{x} + 5}{\frac{x}{x}}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 7.2.2

Rewrite the expression.

$\frac{\sqrt{lim_{x \to \infty} \frac{\frac{6}{x} + 5}{1}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 7.3

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

$\frac{\sqrt{\frac{lim_{x \to \infty} \frac{6}{x} + 5}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 7.4

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

$\frac{\sqrt{\frac{lim_{x \to \infty} \frac{6}{x} + lim_{x \to \infty} 5}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 7.5

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

$\frac{\sqrt{\frac{6 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} 5}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 8

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

$\frac{\sqrt{\frac{6 \cdot 0 + lim_{x \to \infty} 5}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 9

Evaluate the limit.

Tap for more steps...

Step 9.1

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

$\frac{\sqrt{\frac{6 \cdot 0 + 5}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 9.2

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

$\frac{\sqrt{\frac{6 \cdot 0 + 5}{1}}}{lim_{x \to \infty} 6 - \frac{1}{x}}$

Step 9.3

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

$\frac{\sqrt{\frac{6 \cdot 0 + 5}{1}}}{lim_{x \to \infty} 6 - lim_{x \to \infty} \frac{1}{x}}$

Step 9.4

Evaluate the limit of $6$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{\frac{6 \cdot 0 + 5}{1}}}{6 - lim_{x \to \infty} \frac{1}{x}}$

Step 10

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

$\frac{\sqrt{\frac{6 \cdot 0 + 5}{1}}}{6 - 0}$

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

Divide $6 \cdot 0 + 5$ by $1$ .

$\frac{\sqrt{6 \cdot 0 + 5}}{6 - 0}$

Step 11.2

Simplify the numerator.

Tap for more steps...

Step 11.2.1

Multiply $6$ by $0$ .

$\frac{\sqrt{0 + 5}}{6 - 0}$

Step 11.2.2

Add $0$ and $5$ .

$\frac{\sqrt{5}}{6 - 0}$

Step 11.3

Simplify the denominator.

Tap for more steps...

Step 11.3.1

Multiply $- 1$ by $0$ .

$\frac{\sqrt{5}}{6 + 0}$

Step 11.3.2

Add $6$ and $0$ .

$\frac{\sqrt{5}}{6}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5}}{6}$

Decimal Form:

$0.37267799 \dots$