Find the limit Lim [x tends to a] {e ^ a [e ^ (x-a) - 1]} / (x-a)

Find the limit Lim [x tends to a] {e ^ a [e ^ (x-a) - 1]} / (x-a)

lim【x→a】{e^a[e^(x-a)-1]}/(x-a)
=e^a lim【x→a】[e^(x-a)-1]/(x-a)
=e^a lim【x→a】(x-a)/(x-a)
=e^a×1
=e^a

LIM (x tends to 0) [1 / (e ^ x-1) - 1 / x] to find the limit

1/(e^x-1)-1/x
=(x-e^x+1)/x(e^x-1)
=(1-e^x)/(e^x-1+xe^x)
= -e^x/(e^x+e^x+xe^x)
= -1/(x+2)
= -1/2

When Lim SiNx x tends to zero, why can SiNx be reduced to X and sin1 / X not to 1 / x

Because when x goes to zero,
The limit value of SiNx / X is 1, that is to say, SiNx and X are equivalent,
So SiNx can be reduced to X
When x tends to zero, 1 / X tends to infinity,
Obviously, sin1 / X does not tend to infinity, and they are not equivalent
So sin1 / X cannot be changed to 1 / X