Consider the function m(x)= |x|-x e^x-1

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

Problem

Consider the function m(x)=∣x∣−xex−1,x≠0,m(x)=\frac{|x|-x}{e^x-1},\qquad x\ne 0, and define m(0)m(0) so that mm is continuous at x=0x=0. (a) Show that lim⁡x→0−m(x)\displaystyle\lim_{x\to 0^-}m(x) can be expressed as a ratio whose numerator and denominator each approach 00. Compute this limit. (b) Compute lim⁡x→0+m(x)\displaystyle\lim_{x\to 0^+}m(x). (c) Determine the value that must be assigned to m(0)m(0) so that mm is continuous at x=0x=0. Justify your conclusion.

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