Calculus Examples

$lim_{x \to \infty} \frac{\sqrt{9 x^{2} + 4}}{6 x + 1}$

Step 1

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

$lim_{x \to \infty} \frac{\sqrt{\frac{9 x^{2}}{x^{2}} + \frac{4}{x^{2}}}}{\frac{6 x}{x} + \frac{1}{x}}$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{9 x^{2}}{x^{2}} + \frac{4}{x^{2}}}}{6 + \frac{1}{x}}$

Step 2.2

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 2.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{\frac{9 x^{2}}{x^{2}} + \frac{4}{x^{2}}}}{6 + \frac{1}{x}}$

Step 2.2.2

Divide $9$ by $1$ .

$lim_{x \to \infty} \frac{\sqrt{9 + \frac{4}{x^{2}}}}{6 + \frac{1}{x}}$

Step 2.3

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

$\frac{lim_{x \to \infty} \sqrt{9 + \frac{4}{x^{2}}}}{lim_{x \to \infty} 6 + \frac{1}{x}}$

Step 2.4

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to \infty} 9 + \frac{4}{x^{2}}}}{lim_{x \to \infty} 6 + \frac{1}{x}}$

Step 2.5

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

$\frac{\sqrt{lim_{x \to \infty} 9 + lim_{x \to \infty} \frac{4}{x^{2}}}}{lim_{x \to \infty} 6 + \frac{1}{x}}$

Step 2.6

Evaluate the limit of $9$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{9 + lim_{x \to \infty} \frac{4}{x^{2}}}}{lim_{x \to \infty} 6 + \frac{1}{x}}$

Step 2.7

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

$\frac{\sqrt{9 + 4 lim_{x \to \infty} \frac{1}{x^{2}}}}{lim_{x \to \infty} 6 + \frac{1}{x}}$

Step 3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{\sqrt{9 + 4 \cdot 0}}{lim_{x \to \infty} 6 + \frac{1}{x}}$

Step 4

Evaluate the limit.

Tap for more steps...

Step 4.1

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

$\frac{\sqrt{9 + 4 \cdot 0}}{lim_{x \to \infty} 6 + lim_{x \to \infty} \frac{1}{x}}$

Step 4.2

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

$\frac{\sqrt{9 + 4 \cdot 0}}{6 + lim_{x \to \infty} \frac{1}{x}}$

Step 5

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

$\frac{\sqrt{9 + 4 \cdot 0}}{6 + 0}$

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

Multiply $4$ by $0$ .

$\frac{\sqrt{9 + 0}}{6 + 0}$

Step 6.1.2

Add $9$ and $0$ .

$\frac{\sqrt{9}}{6 + 0}$

Step 6.1.3

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}}}{6 + 0}$

Step 6.1.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3}{6 + 0}$

Step 6.2

Add $6$ and $0$ .

$\frac{3}{6}$

Step 6.3

Cancel the common factor of $3$ and $6$ .

Tap for more steps...

Step 6.3.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{6}$

Step 6.3.2

Cancel the common factors.

Tap for more steps...

Step 6.3.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 1}{3 \cdot 2}$

Step 6.3.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 2}$

Step 6.3.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$