Limits at Infinity

Below are a range of examples of limits at x goes to +∞ or -∞. If you need assistanace with the terminology or the general concepts and techniques, follow the relevant links.

1. Compute the limit of the following rational expression as x →∞.

lim x---3- x→ ∞ x2 - 1
2. Compute the limit of the following quotient as x →∞. Note that there are special challenges when roots are involved.
√------- -4x2-+-9- lxi→m∞ 3 - x
3. Compute the limit of the following quotient as x →-∞.
√------- -4x2-+-9- lxi→m∞ 3 - x
4. Compute the limit of the following rational expression as x →∞.

[Hint: The answer will be infinite and the technique will be somewhat different in the past because the relationship between the essential degree of the numerator and denominator.]

x2 + 1 lim ------ x→ ∞ 3 - x
5. Compute the limit of the following quotient as x →∞ using the squeeze theorem.
sin(x) xli→m∞ -2---- x + 1

[Hint: The numerator has no polynomials or roots of polynomials, so the multiplication- division technique used above is not going to help. However -1 ≤ sin(x) ≤ 1 so another route is possible.]