Find the Asymptotes y=( square root of x)/(2x^2-10)

$y = \frac{\sqrt{x}}{2 x^{2} - 10}$

Step 1

Find where the expression $\frac{\sqrt{x}}{2 x^{2} - 10}$ is undefined.

$x < 0, x = \sqrt{5}$

Step 2

Since $\frac{\sqrt{x}}{2 x^{2} - 10}$ $\to$ $- \infty$ as $x$ $\to$ $\sqrt{5}$ from the left and $\frac{\sqrt{x}}{2 x^{2} - 10}$ $\to$ $\infty$ as $x$ $\to$ $\sqrt{5}$ from the right, then $x = \sqrt{5}$ is a vertical asymptote.

$x = \sqrt{5}$

Step 3

Evaluate $lim_{x \to \infty} \frac{\sqrt{x}}{2 x^{2} - 10}$ to find the horizontal asymptote.

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $\sqrt{x^{4}} = x^{2}$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{x}{x^{4}}}}{\frac{2 x^{2}}{x^{2}} + \frac{- 10}{x^{2}}}$

Evaluate the limit.

Simplify each term.

$lim_{x \to \infty} \frac{\sqrt{\frac{x}{x^{4}}}}{2 - \frac{10}{x^{2}}}$

Cancel the common factor of $x$ and $x^{4}$ .

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{x^{1}}{x^{4}}}}{2 - \frac{10}{x^{2}}}$

Factor $x$ out of $x^{1}$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{x \cdot 1}{x^{4}}}}{2 - \frac{10}{x^{2}}}$

Cancel the common factors.

Factor $x$ out of $x^{4}$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{x \cdot 1}{x \cdot x^{3}}}}{2 - \frac{10}{x^{2}}}$

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{\frac{x \cdot 1}{x \cdot x^{3}}}}{2 - \frac{10}{x^{2}}}$

Rewrite the expression.

$lim_{x \to \infty} \frac{\sqrt{\frac{1}{x^{3}}}}{2 - \frac{10}{x^{2}}}$

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

$\frac{lim_{x \to \infty} \sqrt{\frac{1}{x^{3}}}}{lim_{x \to \infty} 2 - \frac{10}{x^{2}}}$

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to \infty} \frac{1}{x^{3}}}}{lim_{x \to \infty} 2 - \frac{10}{x^{2}}}$

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$\frac{\sqrt{0}}{lim_{x \to \infty} 2 - \frac{10}{x^{2}}}$

Evaluate the limit.

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

$\frac{\sqrt{0}}{lim_{x \to \infty} 2 - lim_{x \to \infty} \frac{10}{x^{2}}}$

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

$\frac{\sqrt{0}}{2 - lim_{x \to \infty} \frac{10}{x^{2}}}$

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{0}}{2 - 10 lim_{x \to \infty} \frac{1}{x^{2}}}$

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

$\frac{\sqrt{0}}{2 - 10 \cdot 0}$

Simplify the answer.

Simplify the numerator.

Rewrite $0$ as $0^{2}$ .

$\frac{\sqrt{0^{2}}}{2 - 10 \cdot 0}$

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

$\frac{0}{2 - 10 \cdot 0}$

Simplify the denominator.

Multiply $- 10$ by $0$ .

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

Add $2$ and $0$ .

$\frac{0}{2}$

Divide $0$ by $2$ .

$0$

Step 4

List the horizontal asymptotes:

$y = 0$

Step 5

Use polynomial division to find the oblique asymptotes. Because this expression contains a radical, polynomial division cannot be performed.

Cannot Find Oblique Asymptotes

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = \sqrt{5}$

Horizontal Asymptotes: $y = 0$

Cannot Find Oblique Asymptotes

Step 7

Find the Asymptotes y=( square root of x)/(2x^2-10)

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