Question

Evaluate the Given limit: \(\lim_{x\rightarrow \frac{\pi}{2}}\frac{tan\,2x}{x}-\frac{\pi}{2}\)

Answer 1

Given:

lim_x→π/2 tan 2x / (x − π/2)

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Step 1: Substitute x = π/2 + h

As x → π/2, let
x = π/2 + h,   where h → 0

Then,
2x = π + 2h

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Step 2: Simplify the numerator

tan(2x) = tan(π + 2h)

Using identity,
tan(π + θ) = tan θ

tan(2x) = tan(2h)

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Step 3: Rewrite the limit

lim_h→0 tan(2h) / h

= lim_h→0 2 · [tan(2h) / (2h)]

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Step 4: Use standard limit

lim_θ→0 tan θ / θ = 1

So,

lim_h→0 tan(2h) / h = 2

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Final Answer:

lim_x→π/2 tan(2x) / (x − π/2) = 2