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

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

Sum

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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Chapter 9: Differential Calculus - Limits and Continuity - Exercise 9.1 [Page 98]

Chapter 9 Differential Calculus - Limits and Continuity
Exercise 9.1 | Q 23 | Page 98

Evaluate the following limit:

`lim_(x -> 3)[sqrt(2x + 6)/x]`

Evaluate the following limit:

`lim_(x -> 2)[(x^(-3) - 2^(-3))/(x - 2)]`

Evaluate the following limit:

If `lim_(x -> 1)[(x^4 - 1)/(x - 1)]` = `lim_(x -> "a")[(x^3 - "a"^3)/(x - "a")]`, find all possible values of a

In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.

`lim_(x -> 2) (x^2 - 1)` = 3

Evaluate the following :

`lim_(x -> 1) [(x + 3x^2 + 5x^3 + ... + (2"n" - 1)x^"n" - "n"^2)/(x - 1)]`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> 1) f(x)` where `f(x) = {{:(x^2 + 2",", x ≠ 1),(1",", x = 1):}`

Sketch the graph of a function f that satisfies the given value:

f(0) is undefined

`lim_(x -> 0) f(x)` = 4

f(2) = 6

`lim_(x -> 2) f(x)` = 3