How do I find the limit as 'x' approaches negative infinity?

Question

_{One of my homework problems asks me to find the limit as 'x' approaches negative infinity of} 2x⁷ - 5x³ .

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5x⁷ + 2x⁴

^{How would I solve? Thank you.}

Answer 1

You may rewrite the polynomial as x^7(5+2/x^3). It is obvious that the term inside the parenthesis goes to the number 5 as x goes to -\infty. But the term x^7 goes to -\infty as x goes to -\infty since the power 7 is odd. Hence, the final answer is that the quantity 5x^7+2x^4 goes to -\infty as x goes to -\infty.

Unless you want to find the limit of the expression (2x^7-5x^3)/(5x^7+2x^4) (which is not clear from your statement of the question). In this case the dominating powers are the biggest from the numerator and the denominator which means that the limit as x goes to -\infty will be equal to 2/5.