The following limit
ℓ=limx→0xcosx−sinxx2
is a nice candidate for L'Hopital's Rule. This was given at a school before L'Hopital's Rule was covered. I wonder how we can skip the rule and use basic limits such as:
limx→0sinxx,limx→0cosx−1x2
Answer
We have,
limx→0xcosx−sinxx2=limx→0cosx−1x+limx→0x−sinxx2
=−2limx→0sin2(x2)x+limx→0x−sinxx2
The first limit is zero since limx→0sinxx=1, and,
0≤limx→0x−sinxx2≤limx→0tanx−sinxx2
But,
limx→0tanx−sinxx2=limx→0 (sinx×1−cosxx2cosx)=limx→01−cosxx=0
Thus, by the Squeeze Theorem,
limx→0xcosx−sinxx2=0
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