How do I obtain the limit of $x^4+x^5$ when $x$ tends to $-\infty$

Question

Well that's it.

$$ \lim_{x\to \space-\infty} x^4+x^5 $$

How do I go about dealing with the $\infty-\infty$.

Thanks!

Answer 1

Try factoring out the "dominant term" -- which in this case is $x^5$. By rewriting this as $$\lim_{x \to -\infty} x^5 \left( \frac 1 x + 1 \right)$$ you will have a product of terms that will be more manageable than the $\infty - \infty$ case.

Answer 2

While $\infty-\infty$ is indeed indeterminate, $x^5$ grows in absolute value far faster than $x^4,$ so the limit is simply $\lim_{x\rightarrow -\infty}x^5=-\infty.$

Answer 3

A little more formal:

$\lim _{x \rightarrow -\infty} p(x) := x^4+ x^5 =$

$ x^5(1/x +1)$

Let $y:= - x$; and $P(y):= p(-y)$.

We now consider

$\lim_{ y \rightarrow \infty} P(y):$

$P(y)= -y^5(1-1/y) \lt $

$-y^5 (1/2) \lt -y/2,$ for $y > 2$.

$\lim_{y \rightarrow \infty} P(y):$

For every $2M \in \mathbb{R^+}$, $2M \gt 2$,

and $y \gt 2M$:

$P(y) \lt -y/2 \lt -M.$