Question

Find the limit.

Answer 1

\[\lim_{x \rightarrow \infty} \frac{ 3x^3+2}{ 9x^3-2x^2+7 }\]

Answer 2

\[\frac{3}{9}=\frac{1}{3}\]

Answer 3

if the degrees are the same, it is the ratio of the leading coefficients

Answer 4

What if they're different? for example, \[x/\sqrt{x^2-x}\]

Answer 5

then pretend the denominator is x and do the same thing

Answer 6

so the answer for the second question would be 1?

Answer 7

yes. you can ignore the lower power of x in the denominator and think of this as
\[\frac{x}{\sqrt{x^2}}=\frac{x}{x}=1\] in the limit

Answer 8

this is probably not what your teacher wants you to do, but it is a good way to visualize it

Answer 9

Thank you!

Answer 10

yw
(easy right?)

Answer 11

yeah, now that I looked at it the way you told me to. My teacher just confused me.

Answer 12

probably wanted you to do something annoying and cumbersome like writing
\[\frac{x}{\sqrt{x^2+x}}=\frac{\frac{x}{x}}{\sqrt{\frac{x^2}{x^2}-\frac{x}{x^2}}}\]
\[=\frac{1}{\sqrt{1+\frac{1}{x}}}\] and then take the limit, saying that the second term in the denominator goes to zero