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Grade:11

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Riddhish Bhalodia
askIITians Faculty 434 Points
8 years ago
We start with substituting x = a + t, and hence as x tends to a, t tends to 0. so the limit now becomes
L = lim_{t\rightarrow 0} (-t)tan(\frac{\pi}{2}(1 + \frac{t}{a})) = lim_{t\rightarrow 0} tcot(\frac{\pi t}{2a})
this can be written in form of standard limit multiples as
L = lim_{t\rightarrow 0} cos(\frac{\pi t}{2a})\frac{\frac{\pi t}{2a}}{sin(\frac{\pi t}{2a})} \frac{2a}{\pi}
now as per the standard limit rules the limits of the first two factos is 1 and hence
L = \frac{2a}{\pi}
Proved!

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