You made a mistake there. You've written an expression which divides by zero, which is meaningless. Perhaps you meant sqrt(x e^sin(pi/x)) Finding the limit as x->0 involves the epsilon/delta game. Suppose the limit of the expression is L. You pick a small value delta. You challenge me to find an x which produces an expression which is within delta. I have to produce a related value epsilon, such that for all values of x within epsilon of 0, the expression is within delta of L. If I can do this, then L is the limit of the expression. This expression sqrt(x e^sin(pi/x)) is a nice one for this. I see that sin(pi/x) must be within -1 and 1 so e^sin(pi/x) must be within e^-1 and e^1 In fact it must be within 0 and 3 So x e^sin(pi/x) must be between 0 and 3x and sqrt(x e^sin(pi/x)) must be between 0 and sqrt(3x) (assuming that we're taking the positive root). We want to test if L=0 is the limit of the expression. When you chalenge me to make the expression within delta of L=0, that is, within -delta and delta, I set my epsilon to (delta/3)^2 so that delta = sqrt(3 epsilon) I now know that if x is less than epsilon, then sqrt(3x) is less than delta and I win the game. Therefore L=0 is the limit.