Liimit as x approaches negative infinity

[idllotsaroms]

limit as x approaches negative infinity of \(\displaystyle (x^4 + x^5)\)
Ive factored out \(\displaystyle (x^4)\) to get \(\displaystyle x^4(1 + x)\). However, the answer I seem to be getting is infinity - infinity, when the answer IS -infinity.

Because \(\displaystyle (-infinity)^4 (1) = +infinity\) and \(\displaystyle (-infinity)^4 (-infinity) = -infinity\) thus, I get \(\displaystyle infinity - infinty\)

Can someone please help me out and point out what I've done incorrectly?

 

[DrPhil]

limit as x approaches negative infinity of \(\displaystyle (x^4 + x^5)\)
Ive factored out \(\displaystyle (x^4)\) to get \(\displaystyle x^4(1 + x)\). However, the answer I seem to be getting is infinity - infinity, when the answer IS -infinity.

Because \(\displaystyle (-infinity)^4 (1) = +infinity\) and \(\displaystyle (-infinity)^4 (-infinity) = -infinity\) thus, I get \(\displaystyle infinity - infinty\)

Can someone please help me out and point out what I've done incorrectly?

You have factored out x^4, which you have shown is always positive, no matter what the sign of x.

The other factor is (1 + x), which has the same sign as x when x gets large. The product with x^4 has the same sign as this factor.

The expression \(\displaystyle \infty - \infty\) is indeterminant - there is no way to evaluate it. That is why you had to factor out the x^4 to find the answer.

 

[lookagain]

limit as x approaches negative infinity of \(\displaystyle (x^4 + x^5)\)

Here is a quick rule of thumb. If you have the limit as, say, for x, as x approaches +/- oo of a polynomial (in x) expression, ignore all terms except that of the highest degree.
\(\displaystyle So, \ \ \displaystyle\lim_{x \to \ -oo }(x^4 + x^5) \ = \ \displaystyle\lim_{x \to \ -oo} x^5 \ = \ -oo\)

 

[srmichael]

Here is a quick rule of thumb. If you have the limit as, say, for x, as x approaches +/- oo of a polynomial (in x) expression, ignore all terms except that of the highest degree.
\(\displaystyle So, \ \ \displaystyle\lim_{x \to \ -oo }(x^4 + x^5) \ = \ \displaystyle\lim_{x \to \ -oo} x^5 \ = \ -oo\)

Lookagain is correct, because eventually as the x gets to be extremely large, the value of the term with the highest degree will eventually dominate over all the other terms. All the other terms will be infitesimal in value as x approaches infinity....like spitting in the ocean and seeing how much the level rises.