Tamil Nadu Board of Secondary EducationHSC Commerce Class 11th

If f(x)= {x-|x|xif x≠02ifx=0 then show that limx→1f(x) does not exist. - Business Mathematics and Statistics

Sum

If f(x)= {((x - |x|)/x if  x ≠ 0),(2 if x = 0):} then show that lim_(x->1)f(x) does not exist.

Solution

"L"["f"(x)]_(x=0) = lim_(x->0^-) "f"(x) = lim_(h->0) "f"(0 - "h")

= lim_(h->0) "f"(-"h") = lim_(h->0) ((-"h") - |- "h"|)/(-"h")

= lim_(h->0) (- "h" - "h")/(- "h")

= lim_(h->0) (-2cancel("h"))/(-cancel(h))

= lim_(h->0) 2 = 2  ...[∵ |- h| = h]   ...(1)

"R"["f"(x)]_(x=0) = lim_(x->0) "f"(x) = lim_(x->0) "f"(0 + "h")

= lim_(h->0) f(h)

= lim_(h->0) ("h" - |"h"|)/"h"

= lim_(h->0) ("h - h")/"h"

= lim_(h->0) 0/"h" = 0   ...(2)

From (1) and (2),

"L"["f"(x)]_(x=0) ne "R"["f"(x)]_(x=0)

therefore lim_(x-> ∞) "f"(x) does not exist.

Concept: Limits and Derivatives
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Tamil Nadu Board Samacheer Kalvi Class 11th Business Mathematics and Statistics Answers Guide
Chapter 5 Differential Calculus
Miscellaneous Problems | Q 3 | Page 125
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