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(lim)(x->pi/2)(tan2x)/(x-pi/2)...

`(lim)_(x->pi/2)(tan2x)/(x-pi/2)`

Text Solution

Verified by Experts

`underset(xto(pi)/2)(lim)(tan2x)/(x-pi/2)`


LHL `= underset(hto 0^-)(lim)(tan2(pi/2-h))/(pi/2-h-pi/2)`

` " " " " =underset(hto 0^-)(lim)(tan(pi-2h))/(-h)`

` " " " " =underset(hto 0^-)(lim)(-tan2h)/(-h)`

` " " " " =underset(hto 0^-)(lim)(-2tan2h)/(-2h)=2`


RHL `= underset(hto 0^+)(lim)(tan2(pi/2+h))/(pi/2+h-pi/2)`

` " " " " = underset(hto 0^+)(lim)(tan(pi+2h))/(h)`

` " " " " = underset(hto 0^+)(lim)(tan2h)/(h)`

` " " " " = underset(hto 0^+)(lim)(2tan2h)/(2h)=2`

Thus,
`underset(xto(pi)/2)(lim)(tan2x)/(x-pi/2)=2`
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