The limit represents the derivative of some function *f*(*x*) at some number *a*. Select an appropriate *f*(*x*) and *a*

(tan (x) - 1)/(x-pi/4)

Lim -> pi/4

Pick all that match:

*A) f*(*x*) = tan(x), *a* = *π*/4

B) *f*(*x*) = tan(*x*), *a* = *π*

*C) f(x) = tan(x) - 1, a = π*

*D) f(x) = tan(x), a = 1/4*

*E) f(x) = tan(x) - 1, a = 1/4*

*F) None of the Above*

### 1 Answer | Add Yours

Limit (tanx -1)/(x-pi/2) as x-->pi/2.

The definition of d/dx{tanx) = Lt {tanx(x+h) -tanx) }/{(x+h)-x} as h--> 0

= Lt{(tanx +tanh)/(1-tanx*tanh) -tanx)}/h as --> 0

=Lt (tanx+tanh- tanx +tan^2x*tanh)/{h(1-tanx*tanh)} as h-->0

= Lt (tanh)(1+tan^x)/{h(1-tanx*tanh) as h-->0

= {1}{(1+tan^x)/1} as Lt tax/h = 1 as x-->0. And lt tanx*tanh = 0 as h-->0

= sec^2x.

Therefore Lt {tanx -1}/(x-pi/4) as x--> pi/4 is equal to {d/dx(tanx) at x = pi/4} = {secant (pi/2}^2 = (sqrt2)^2 = 2.

So f(x) = tanx

And {d/dx(f(x) at x= pi/4} = {d/dx (tanx at x= pi/4} = (secant pi/4)^2 = 2 is correct and the choice at a is nearl correct but confusing the young mind.

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