Question

Find \(\lim_{x\rightarrow 0}\) f(x) and \(\lim_{x\rightarrow 1}\) f(x) where { 2x+3, x≤0 3 (x+1), x>0

Answer 1

Given:

f(x) =
2x + 3,   x ≤ 0
3(x + 1),   x > 0

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Part (i): Evaluate lim_x→0 f(x)

Step 1: Left Hand Limit (LHL)

LHL = lim_x→0⁻ f(x)

= lim_x→0⁻ (2x + 3)

= 3

Step 2: Right Hand Limit (RHL)

RHL = lim_x→0⁺ f(x)

= lim_x→0⁺ 3(x + 1)

= 3

Step 3: Compare LHL and RHL

Since LHL = RHL = 3,

lim_x→0 f(x) = 3

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Part (ii): Evaluate lim_x→1 f(x)

For x → 1, we have x > 0,

So, f(x) = 3(x + 1)

lim_x→1 f(x) = 3(1 + 1)

= 6

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Final Answer:

lim_x→0 f(x) = 3
lim_x→1 f(x) = 6