Let the function be defined near by the ratio F(x)= p(x) q(x)

MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

Let the function FF be defined near x=2x=2 by the ratio F(x)=p(x)q(x),F(x)=\frac{p(x)}{q(x)}, where p(x)={x2−4,x<2,sin⁡(π(x−2)),x>2,q(x)={x−2,x<2,ln⁡x−ln⁡2,x>2.p(x)=\begin{cases}x^2-4,&x<2,\\ \sin\bigl(\pi(x-2)\bigr),&x>2,\end{cases}\qquad q(x)=\begin{cases}\sqrt{x}-\sqrt{2},&x<2,\\ \ln x-\ln 2,&x>2.\end{cases} Answer the following questions about the limit of F(x)F(x) as xx approaches 22. (a) Show that lim⁡x→2−F(x)\displaystyle\lim_{x\to 2^-}F(x) has the algebraic form 00\dfrac{0}{0}. State explicitly the limiting values of the numerator and denominator that produce this form. (b) Explain why a direct substitution of x=2x=2 into the ratio is insufficient to evaluate lim⁡x→2−F(x)\displaystyle\lim_{x\to 2^-}F(x) and describe, in general terms, what further analysis is necessary to resolve the limit. (c) Verify that the limit in part (a) exists and determine its value. Provide full justification for each step of your work.

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