Question

Lt x approach 0 (e^x-e-^-x/sinx)​

Answer 1

lim(x,y)→(z,z)f(x)−f(y)x−y=f′(z) or well behaved f.

n our case,  f  (x)=ex  and  z=0  (which both  x  and  sin(x)  approach as  x  approaches  0 ), so our answer is  f  ′(0)=e0=1 .

[A note on “well-behaved” for such folks as would care about such things: This “strengthened derivative” property automatically holds whenever  f  is differentiable in the standard slightly weaker sense throughout an interval around  z , with its derivative continuous at  z . Also, whenever the left-hand side is well-defined, so is the right-hand side and they match; the only possibility of mismatch is that sometimes the right-hand side is well-defined while the left-hand side isn’t. Basically, you will always have this equation in elementary calculus working only with smooth functions, unless you are very specifically seeking out pathological functions which lack it.]