Limits at Infinity

Question

What is the limit at infinity and negative infinity of this function:

f(t)=e^t - t

Show steps so I can understand how it can be done.

Answer 1

Write f(t)=e^t - t as f(t)= t (e^t / t - 1).

As t goes to +infinity, the first factor t goes to +infinity and the second factor (e^t / t - 1) goes to +infinity. (To see why this holds for the second factor, note that e^t grows much faster than t, or just apply L'Hopital's rule). Hence, f(t) goes to +infinity as t goes to +infinity.

As t goes to -infinity, the first factor t goes to -infinity and the second factor (e^t / t - 1) goes to -1 (because e^t goes to 0, and so does e^t / t). Hence, f(t) goes to +infinity as t goes to +infinity.