Verify the existence of limx→1f(x), where ,for,forf(x)={|x-1|x-1, for x≠10, for x=1 - Mathematics

Verify the existence of `lim_(x -> 1) f(x)`, where `f(x) = {{:((|x - 1|)/(x - 1)",", "for" x ≠ 1),(0",", "for" x = 1):}`

बेरीज

Given `f(x) = {{:((|x - 1|)/(x - 1)",", "for" x ≠ 1),(0",", "for" x = 1):}`

`f(x) = {{:(|(x - 1|)/(x - 1), "for" x < 1 and x > 1),(0, "for" x = 1):}`

`f(x) = {{:((- (x - 1))/(x - 1), "for" x < 1),((x - 1)/(x - 1), "for" x > 1),(0, "for" x = 1):}`

`f(x) = {{:(-1, "for" x < 1),(1, "for" x > 1),(0, "for" x = 1):}`

`f(1^-) = lim_(x -> 1^-) f(x)`

= `lim_(x -> 1^-) (- 1)` = – 1 .......(1)

`f(1^+) = lim_(x -> 1^+) f(x)`

= `lim_(x -> 1^+) (1)` = 1 .......(2)

From equations (1) and (2) we get

f(1^–) ≠ f(1⁺)

∴ The limit of f(x) does not exist.

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पाठ 9: Differential Calculus - Limits and Continuity - Exercise 9.1 [पृष्ठ ९८]

पाठ 9 Differential Calculus - Limits and Continuity
Exercise 9.1 | Q 23 | पृष्ठ ९८

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