Consider the function

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Consider the function f(x)=x−3x−9,x≠9f(x)=\dfrac{\sqrt{x}-3}{x-9},\quad x\neq 9. The graph of y=f(x)y=f(x) is shown to be continuous for x≠9x\neq 9. (a) Determine lim⁡x→9−f(x)\displaystyle\lim_{x\to 9^{-}}f(x). (b) Determine lim⁡x→9+f(x)\displaystyle\lim_{x\to 9^{+}}f(x) and state whether the two one-sided limits from parts (a) and (b) agree. (c) Verify that your answers in parts (a) and (b) are consistent with the definition of the derivative of x\sqrt{x} at x=9x=9. Explain your reasoning.

Related theory

Browse the problem library