Find lim as x goes to 0 of (xsinx)/(1-cosx)
To find the limit as x approaches 0 of (xsinx)/(1-cosx), we can use algebraic manipulation and apply L'Hôpital's rule if necessary. Here's how you can proceed:
Step 1: Simplify the expression by multiplying the numerator and denominator by (1+cosx):
lim(x->0) [(xsinx)/(1-cosx)] = lim(x->0) [(xsinx)(1+cosx)] / [(1-cosx)(1+cosx)].
Step 2: Use the identity sin²x = 1 - cos²x to rewrite the numerator:
lim(x->0) [x(1-cos²x)(1+cosx)] / [(1-cosx)(1+cosx)].
Step 3: Cancel out the common factors (1-cosx) from the numerator and denominator:
lim(x->0) x(1+cosx) / (1+cosx).
Step 4: Now, we can see that (1+cosx) cancels out in the numerator and denominator:
lim(x->0) x.
Step 5: Finally, evaluate the limit as x approaches 0:
lim(x->0) x = 0.
Therefore, the limit as x goes to 0 of (xsinx)/(1-cosx) is equal to 0.