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
Steve M
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 .

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