Question

!!!!! determine the function being differentiated, and the number at which its derivative is being evaluated. Where possible, evaluate the limits using differentiation.

Answer 1

Recall that the derivative of a function f(x) at a point x = c is given by

$\displaystyle f'(c) = \lim_{x\to c} \frac{f(x) - f(c)}{x - c}$

By substituting h = x - c, we have the equivalent expression

$\displaystyle f'(c) = \lim_{h\to0} \frac{f(c+h) - f(c)}h$

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,

$\displaystyle \lim_{h\to0} \frac{\sin\left(\frac\pi3 + h\right) - \frac{\sqrt3}2}h$

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

$(\sin(x))' = \cos(x)$

and cos(π/3) = 1/2.

For the second limit,

$\displaystyle \lim_{a\to0} \frac{e^{2a} - 1}a$

we observe that e²ˣ = 1 if x = 0. So this limit is the derivative of e²ˣ at x = 0. We have

$\left(e^{2x}\right)' = e^{2x} (2x)' = 2e^{2x}$

and 2e⁰ = 2.