The limit of e^(1/x) as x approaches 0 depends on the direction of the approach. When approaching from the left (b -> 0-), 1/b tends to negative infinity, leading to the limit of e^(1/x) approaching 0. Conversely, when approaching from the right (x -> 0+), the limit diverges to positive infinity, as indicated by the expression lim_{x -> 0+} e^(1/x) = +∞. The discussion highlights the importance of direction in evaluating limits involving exponential functions.
PREREQUISITESStudents of calculus, mathematics educators, and anyone interested in understanding the nuances of limits and exponential functions.
b approaches 0 from below, so 1/b tends to minus infinity. There is a small mistake: the 1 in the (1 . 0 - 0) term. Need to use the fact that x e-x tends to 0 as x tends to +infinity.Phys12 said:
In this solution, in the last 3rd line, I get the first part (-e^-1 - e^-1), however, after the '-' symbol, the person writes (1/b * e^1/b - e^1/b) and takes the limit as b->0. However, shouldn't this give him (inf. * e^inf - e^inf)?
Thanks
(His answer is correct, by the way)
Ohk! I didn't notice the fact that we were approaching 0 from the left hand side. Thanks! :)haruspex said:
Phys12 said: