The limit represents f'(c) for a function f(x) and a number c. Find f(x) and c.
lim_∆→0 [3 - 2(9 + Δx) - (-15)]/Δx

Solution:

Given: f'(x ) = [3 - 2(9 + Δx) - (-15)]/Δx

We are given the limit f'(c).

We know that f'(x) = lim_∆→0 [f(x + Δx) - f(x)]/ Δx ---->(1)

So, f'(c) = lim_∆→0 [f(x + Δx) - f(x)]/ Δx

Here, f'(c) = lim_∆→0 [f(3-18+ Δx) - (-15)]/ Δx

f'(c) = lim_∆→0 [f(Δx - 15) - (-15)]/ Δx ---->(2)

Comparing (1) and (2), we get

c = -15 and f(x) = x

⇒ f(-15) = -15

⇒ f'(x ) = 1

Therefore, the values of f(x) and c are x and -15.

The limit represents f'(c) for a function f(x) and a number c. Find f(x) and c.
lim_∆→0 [3 - 2(9 + Δx) - (-15)]/Δx

Summary:

The limit, lim_∆→0 [3 - 2(9 + Δx) - (-15)]/Δx, representing f'(c) for a function f(x) and a number c, f(x) and c are ‘x’ and -15 respectively.