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2 answers:
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Recall that the derivative of a function f(x) at a point x = c is given by
By substituting h = x - c, we have the equivalent expression
since if x approaches c, then h = x - c approaches c - c = 0.
The two given limits strongly resemble what we have here, so it's just a matter of identifying the f(x) and c.
For the first limit,
recall that sin(π/3) = √3/2. Then c = π/3 and f(x) = sin(x), and the limit is equal to the derivative of sin(x) at x = π/3. We have
and cos(π/3) = 1/2.
For the second limit,
we observe that e²ˣ = 1 if x = 0. So this limit is the derivative of e²ˣ at x = 0. We have
and 2e⁰ = 2.