Limits at Infinity

Question

Theresa is investigating the graph of h(x) = x^2/sinx for her calculus II course. She keeps getting her answer wrong, and doesn’t understand why this would be true, since she’s basing her information off of the following graph:

A. The graph indicates that like the graph of y=x^3, lim x-> oo h(x) = oo and that lim x-> -oo h(x) = -oo Explain why this cannot be true.

 

B. Paste a link that is a more accurate rendition of the graph, which might help Theresa learn more about the function

Answer 1

A. Forget about the graph of y=x³ because you're asked to consider y = x²/sinx.  As x goes toward positive infinity, the numerator approaches positive infinity, but the denominator fluctuates wildly; sometimes sine values are negative, sometimes they're positive, and occasionally they're zero, but their absolute values are never more than 1. Therefore there is no exact tendency that dominates; answers vary between positive infinity and negative infinity, so there's no single answer.

 

Reasoning it out yourself is the best way. No need to "paste a link!"