# 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/4Pick all that match: A) f(x) = tan(x), a...

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), aπ/4

B) f(x) = tan(x), aπ

C) f(x) = tan(x) - 1, aπ

D) f(x) = tan(x), a = 1/4

E) f(x) = tan(x) - 1, a = 1/4

F) None of the Above

neela | High School Teacher | (Level 3) Valedictorian

Posted on

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.