For all , a function is defined analytically by f(x)= (x^ 2 -4)(x-5) |x-2|(x+3) .…

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

For all

x\neq2

, a function

f

is defined analytically by

f(x)=\frac{(x^{2}-4)(x-5)}{|x-2|(x+3)}.

Answer the following related questions. (a) Calculate

\displaystyle \lim_{x\to2^-}f(x)

. Show all algebraic steps and clearly indicate how you address any indeterminate forms. (b) Calculate

\displaystyle \lim_{x\to2^+}f(x)

, providing complete justification of each manipulation. (c) Does

\displaystyle \lim_{x\to2}f(x)

exist? State your conclusion and support it using the results from parts (a) and (b). (d) Classify the type of discontinuity that

f

exhibits at

x=2

and justify your classification with reference to both the one-sided limits and the explicit form of

f(x)