Calculus Examples

$lim_{x \to \infty} \frac{1}{x} \sqrt{3 + 4 x^{2}}$

Step 1

Evaluate the limit.

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Step 1.1

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

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Step 1.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{\frac{3}{x^{2}} + \frac{4 x^{2}}{x^{2}}}}{1}$

Step 1.1.2

Divide $4$ by $1$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{3}{x^{2}} + 4}}{1}$

Step 1.2

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

$\frac{lim_{x \to \infty} \sqrt{\frac{3}{x^{2}} + 4}}{lim_{x \to \infty} 1}$

Step 1.3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to \infty} \frac{3}{x^{2}} + 4}}{lim_{x \to \infty} 1}$

Step 1.4

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

$\frac{\sqrt{lim_{x \to \infty} \frac{3}{x^{2}} + lim_{x \to \infty} 4}}{lim_{x \to \infty} 1}$

Step 1.5

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

$\frac{\sqrt{3 lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} 4}}{lim_{x \to \infty} 1}$

Step 2

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

$\frac{\sqrt{3 \cdot 0 + lim_{x \to \infty} 4}}{lim_{x \to \infty} 1}$

Step 3

Evaluate the limit.

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Step 3.1

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

$\frac{\sqrt{3 \cdot 0 + 4}}{lim_{x \to \infty} 1}$

Step 3.2

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

$\frac{\sqrt{3 \cdot 0 + 4}}{1}$

Step 3.3

Simplify the answer.

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Step 3.3.1

Divide $\sqrt{3 \cdot 0 + 4}$ by $1$ .

$\sqrt{3 \cdot 0 + 4}$

Step 3.3.2

Multiply $3$ by $0$ .

$\sqrt{0 + 4}$

Step 3.3.3

Add $0$ and $4$ .

$\sqrt{4}$

Step 3.3.4

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

$\sqrt{2^{2}}$

Step 3.3.5

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

$2$