Question

find all vertical asymptotes and horizontal asymptotes of the graph of the function.
y=7x/the square root ofx^2+10

Answer 1

\[y = \frac{7x}{\sqrt{x^{2}+10}}\]?

Answer 2

yup.

Answer 3

Well the vertical asymptotes are where the denominator is 0, so solve it for 0:\[\sqrt{x^{2}+10} = 0\]\[x^{2}+10 = 0\]\[x^{2} = -10\] No real value for x will cause the denominator to be 0 so it has no vertical asymptotes.

For horizontal asymptotes, you need to look at the end behavior of the function. That is, when x approaches positive or negative infinity:\[\lim_{x \rightarrow \infty}\frac{7x}{\sqrt{x^{2}+10}}\] Multiply the top and bottom by the denominator to get rid of the radical:\[\lim_{x \rightarrow \infty}\frac{7x\sqrt{x^{2}+10}}{x^{2}+10} \]I don't quite remember exactly how to formally do the rest, but you can see that the +10s are trivial when approaching infinity, so you can just ignore them, so you get:\[\lim_{x \rightarrow \infty}\frac{7x\sqrt{x^{2}}}{x^{2}} = \lim_{x \rightarrow \infty}\frac{7x^{2}}{x^{2}} = 7\]