Question

1)lim [(x+y)sec(x+y)-xsecx]/ y
x-o
2)show that lim |x-4|/x-4 does not exists
x-4
3)lim [1-sinx/2]/[cosx/2{(cosx/4)-
x-22/7 (sinx/4)}]

Answer 1

#1: If you mean x-->0, then take a look at what that leaves:

y secy / y = secy

#2: |x-4| = x-4 if x > 4
|x-4| = -(x-4) if x < 4

so, the limit from the left and right are different.

#3:
lim x-->pi
[1-sinx/2]/[cosx/2{(cosx/4)-(sinx/4)}]
what if x = pi? we have
[1-1]/[(0)(0)] = 0/0, so let's try lHospital's Rule:

-1/2 cos x/2
-----------------------
1/2√2 (sin(pi-x)/4) (3sin x/2 + 1)

-1/2
------------
1/2√2 (0)(4)

= -oo