Define three functions as follows: • , •

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

Problem

Define three functions as follows: •

h(x)=\tan\bigl(x-\tfrac{\pi}{4}\bigr)

, •

g(y)=|y|

, and •

f(z)=\sqrt{1+z}

, whose domain is

z\ge -1

. Let the composite function be

P(x)=f\bigl(g\bigl(h(x)\bigr)\bigr)=\sqrt{1+|\tan(x-\tfrac{\pi}{4})|}

. Investigate the behavior of

P

x

approaches

\tfrac{\pi}{4}

from both sides. (a) Evaluate

\displaystyle \lim_{x\to (\pi/4)^{-}}h(x)

and

\displaystyle \lim_{x\to (\pi/4)^{+}}h(x)

. Show all algebraic and trigonometric reasoning. (b) Using your results from part (a), compute the corresponding one-sided limits of

g\bigl(h(x)\bigr)

x

approaches

\tfrac{\pi}{4}

from the left and the right. (c) Determine the one-sided limits of

P(x)

x

approaches

\tfrac{\pi}{4}

from each side, fully justifying the passage of limits through each layer of the composition. (d) Decide whether

\displaystyle \lim_{x\to \pi/4}P(x)

exists. Provide a thorough justification that references your earlier work and addresses the role of continuity for each function involved.