Calculus Examples

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

Step 1

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

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

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Simplify each term.

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide $4$ by $1$ .

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

Step 2.3

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

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

Step 2.4

Move the limit under the radical sign.

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

Step 2.5

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

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

Step 2.6

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

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

Step 3

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

$\frac{\sqrt{4 + 0}}{lim_{x \to \infty} 1 - \frac{2}{x}}$

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 $\infty$ .

$\frac{\sqrt{4 + 0}}{lim_{x \to \infty} 1 - lim_{x \to \infty} \frac{2}{x}}$

Step 4.2

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

$\frac{\sqrt{4 + 0}}{1 - lim_{x \to \infty} \frac{2}{x}}$

Step 4.3

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

$\frac{\sqrt{4 + 0}}{1 - 2 lim_{x \to \infty} \frac{1}{x}}$

Step 5

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

$\frac{\sqrt{4 + 0}}{1 - 2 \cdot 0}$

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

Add $4$ and $0$ .

$\frac{\sqrt{4}}{1 - 2 \cdot 0}$

Step 6.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2}}}{1 - 2 \cdot 0}$

Step 6.1.3

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

$\frac{2}{1 - 2 \cdot 0}$

Step 6.2

Simplify the denominator.

Tap for more steps...

Step 6.2.1

Multiply $- 2$ by $0$ .

$\frac{2}{1 + 0}$

Step 6.2.2

Add $1$ and $0$ .

$\frac{2}{1}$

Step 6.3

Divide $2$ by $1$ .

$2$