Differentiation and Limit

Calculus Level 2

Function $f(x)$ is differentiable for all real numbers $x$ and satisfies $\displaystyle \lim_{x \to \infty} f'(x)=3.$ What is the value of $\lim_{x \to \infty} (f(x+3)-f(x-3))?$

18 24 12 6

1 solution

Vincent Moroney
Jun 10, 2018

The behavior of the derivative at infinity implies that the original function is linear, i.e in the form of $f(x) = f'(x)x + f(0)$ where $f'(x)=3$ . So we have $\lim_{x\to\infty} 3(x+3) + f(0) - [3(x-3)+f(0) ]$ $\lim_{x\to\infty}18 = \boxed{18}$

Differentiation and Limit

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