Let the function be defined for all by

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

Problem

Let the function gg be defined for all x>−1x>-1 by g(x)=4+x−2xg(x) = \dfrac{\sqrt{4+x}-2}{x}. As xx approaches 00, both the numerator and denominator of g(x)g(x) approach 00. (a) Identify the indeterminate form that arises when x→0+x\to 0^{+} and explain why it is classified as that form. (b) Using appropriate limit theorems, explain why the one-sided limit lim⁡x→0+g(x)\lim_{x\to 0^{+}}g(x) must exist and be finite, even though direct substitution yields 0/00/0. (c) Without carrying out an explicit algebraic simplification, determine the sign (positive or negative) of g(x)g(x) for values of xx that are (i) small and positive and (ii) small and negative (where the expression is defined). Use your reasoning to discuss what this implies about the left-hand and right-hand limits at x=0x=0.

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