Question

Evaluate \(\lim_{x\rightarrow 0}\)f(x), where { |x|/x, x≠0 0, x=0

Answer 1

Given:

f(x) =
|x| / x,   x ≠ 0
0,   x = 0

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Step 1: Evaluate Left Hand Limit (LHL)

For x → 0⁻, x is negative, so

|x| = −x

|x| / x = −1

lim_x→0⁻ f(x) = −1

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Step 2: Evaluate Right Hand Limit (RHL)

For x → 0⁺, x is positive, so

|x| = x

|x| / x = 1

lim_x→0⁺ f(x) = 1

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Step 3: Compare LHL and RHL

LHL ≠ RHL

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Final Answer:

Since left hand limit is not equal to right hand limit,
lim_x→0 f(x) does not exist