Question

Find \(\lim_{x\rightarrow 1}\) f(x) where { x² -1, x≤1 -x²-1, x>1

Answer 1

Given:

f(x) =
x² − 1,   x ≤ 1
−x² − 1,   x > 1

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Step 1: Evaluate Left Hand Limit (LHL)

LHL = lim_x→1⁻ f(x)

= lim_x→1⁻ (x² − 1)

= 1 − 1

= 0

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Step 2: Evaluate Right Hand Limit (RHL)

RHL = lim_x→1⁺ f(x)

= lim_x→1⁺ (−x² − 1)

= −1 − 1

= −2

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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→1 f(x) does not exist