Question

what is the limit of (x^5 + x^2)/(x^3 + x + 1) as x approaches infinity.

please show all work.
thanks in advance

Answer 1

\[\lim_{x \rightarrow \infty}\frac{x^5 +x^2}{x^3+x+1}\]Divide both numerator and denominator by \(x^3\)\[=\lim_{x \rightarrow \infty}\frac{\frac{x^5 +x^2}{x^3}}{\frac{x^3+x+1}{x^3}}\]\[=\lim_{x \rightarrow \infty}\frac{x^2 +\frac{1}{x}}{1+\frac{1}{x^2}+\frac{1}{x^3}}\]When x tends to infinity, 1/x terms would become 0. Can you do it from here?

Answer 2

We have
\[\lim_{x\to \infty}\frac{x^5+x^2}{x^3+x+1}\]

divide numerator and denominator by the highest power of x in denominator, which is x^3
\[\lim_{x\to \infty}\frac{x^5/x^3+x^2/x^3}{x^3/x^3+x/x^3+1/x^3}\]
we get
\[\lim_{x\to \infty}\frac{x^2+1/x}{1+1/x^2+1/x^3}\]
now as
\[x\to \infty, 1/x, 1/x^2, 1/x^3 \to 0\]
so we get
\[\lim_{x\to \infty}\frac{x^2+0}{1+0+0}\ \ \ \ \ \ \ \ \ \ =\ \ \ \ \ \lim_{x\to \infty} x^2\]

Answer 3

rolypoly i dont know how to do complete, can you help me out pls?

Answer 4

as \(x\to \infty\), so what would be the value of x?

Answer 5

is it infinity?

Answer 6

yes the limit is positive infinity....in general if the degree on top is greater than degree on bottom, then limit will go to either +- infinity