How do you prove this is true using the identity (cos(x)-1)/x = 0 as x approaches 0?
#lim x-> 0# #(secx-1) /x# =0
1 Answer
Feb 1, 2018
We seek the limit:
# L = lim_(x rarr 0) (secx-1) /x #
We can write this as:
# L = lim_(x rarr 0) (1/cosx-1) /x #
# \ \ = lim_(x rarr 0) ((1-cosx)/cosx) /x #
# \ \ = lim_(x rarr 0) (1-cosx) /(xcosx) #
# \ \ = lim_(x rarr 0) (cosx-1) /(x) * (-1)/cosx #
# \ \ = {lim_(x rarr 0) (cosx-1) /(x) }{ lim_(x rarr 0) (-1)/cosx} #
And we can evaluate these limits separately:
# lim_(x rarr 0) 1/cosx = 1# and# lim_(x rarr 0) (cosx-1) /(x) = 0 \ \ # (given)
Leading to:
# L = 0 xx (-1) = 0 \ \ # QED
We also note that the given limits is just that ie, it is a limit that is evaluated, rather than an identity .