# trig limits

• Apr 17th 2011, 10:40 AM
frankinaround
trig limits
limit t --> 0 of : (tan(pie/4 + t) - tan(pie/4)) / t

I convert to sins and cosines, try to evaluate and get 1 sometimes, 0 other times. I try to expand out the sin and cos of (pie/4+t) but still get the same answers. I try to multiply both sides by sin(T)/sin(T) and t/t but its messed up because sin(T)= 0 so I usually end up with 1 on the bottom making either the whole expression void or it ends up as sqrt(2)/2 / sqrt(2)/2 which is 1.

So can anyone help? im probably making some big mistakes here, but i mean I think ive been at it for a few hours already......

also the test says the answer = 2
• Apr 17th 2011, 11:22 AM
e^(i*pi)
Have you tried L'hopital's rule?
• Apr 17th 2011, 11:32 AM
topsquark
Quote:

Originally Posted by frankinaround
limit t --> 0 of : (tan(pie/4 + t) - tan(pie/4)) / t

(Crying) By the way, you spell it "pi."

-Dan
• Apr 17th 2011, 12:36 PM
Soroban
Hello, frankinaround!

Quote:

. . . . . . tan(π/4 + t) - tan(π/4)
. . lim .---------------------------
. t →0 . . . . . . . t

. . . . . . . . . . . . . .tan(π/4) + tan(t)
The numerator is: .--------------------- .- .tan(π/4)
. . . . . . . . . . . . . .1 - tan(π/4)tan(t)

. . . . . . . . . . . . . . 1 + tan(t)
. . . . . . . . . . . = . ------------- .- .1
. . . . . . . . . . . . . . 1 - tan(t)

. . . . . . . . . . . . . . 2 tan(t)
. . . . . . . . . . . = . ------------
. . . . . . . . . . . . . . 1 - tan(t)

. . . . . . . . . . . . . . . sin(t)
. . . . . . . . . . . . . . 2 -------
. . . . . . . . . . . . . . . cos(t)
. . . . . . . . . . . = .--------------
. . . . . . . . . . . . . . . .sin(t)
. . . . . . . . . . . . . 1 - -------
. . . . . . . . . . . . . . . .cos(t)

Multiply numerator and denominator by cos(t):

. . . . .2 sin(t)
. . ------------------
. . .cos(t) - sin(t)

. . . . . . . . . . . . .2 sin(t). . . . . . . . . . . . 2. . . . . . sin(t)
Divide by t: . -------------------- . = . ---------------- . --------
. . . . . . . . . . t[cos(t) - sin(t)]. . . . cos(t) - sin(t). . . .t

. . . . . . . . . . . . . . . 2 . . . . . . sin(t) . . . . . 2
Therefore: .lim .---------------- . ------- . = . ------ . 1 . = . 2
. . . . . . . . t→0.cos(t) - sin(t) . . . t . . . . . 1 - 0

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