Calculus Examples

$lim_{x \to \infty} \frac{\sqrt{x^{2} + 3}}{2 x + 1}$

Step 1

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

$lim_{x \to \infty} \frac{\sqrt{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}}{\frac{2 x}{x} + \frac{1}{x}}$

Step 2

Evaluate the limit.

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Step 2.1

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}}{2 + \frac{1}{x}}$

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}}{2 + \frac{1}{x}}$

Step 2.2.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\sqrt{1 + \frac{3}{x^{2}}}}{2 + \frac{1}{x}}$

Step 2.3

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

$\frac{lim_{x \to \infty} \sqrt{1 + \frac{3}{x^{2}}}}{lim_{x \to \infty} 2 + \frac{1}{x}}$

Step 2.4

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to \infty} 1 + \frac{3}{x^{2}}}}{lim_{x \to \infty} 2 + \frac{1}{x}}$

Step 2.5

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

$\frac{\sqrt{lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{3}{x^{2}}}}{lim_{x \to \infty} 2 + \frac{1}{x}}$

Step 2.6

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

$\frac{\sqrt{1 + lim_{x \to \infty} \frac{3}{x^{2}}}}{lim_{x \to \infty} 2 + \frac{1}{x}}$

Step 2.7

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

$\frac{\sqrt{1 + 3 lim_{x \to \infty} \frac{1}{x^{2}}}}{lim_{x \to \infty} 2 + \frac{1}{x}}$

Step 3

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

$\frac{\sqrt{1 + 3 \cdot 0}}{lim_{x \to \infty} 2 + \frac{1}{x}}$

Step 4

Evaluate the limit.

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Step 4.1

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

$\frac{\sqrt{1 + 3 \cdot 0}}{lim_{x \to \infty} 2 + lim_{x \to \infty} \frac{1}{x}}$

Step 4.2

Evaluate the limit of $2$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{1 + 3 \cdot 0}}{2 + lim_{x \to \infty} \frac{1}{x}}$

Step 5

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

$\frac{\sqrt{1 + 3 \cdot 0}}{2 + 0}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $3$ by $0$ .

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

Step 6.1.2

Add $1$ and $0$ .

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

Step 6.1.3

Any root of $1$ is $1$ .

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

Step 6.2

Add $2$ and $0$ .

$\frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$