Evaluate the following limits. \(\displaystyle\lim_{{\theta\rightarrow{0}}}{\frac{{{\sec{\theta}}-{1}}}{{\theta}}}\)

Marzadri9lyy 2022-03-28 Answered
Evaluate the following limits.
limθ0secθ1θ
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Answers (1)

cineworld93uowb
Answered 2022-03-29 Author has 16 answers
We have to find the limit:
limθ0secθ1θ
We know the identity,
cosθsecθ=1
secθ=1cosθ
Substituting above value in the limit,
limθ0secθ1θ=1cosθ1θ
=limθ01cosθcosθθ
=limθ01cosθθcosθ
We have identity,
cos2xsin2x=cos2x
1sin2xsin2x=cos2x
1cos2x=2sin2θ2
then for 1cos2x=2sin2θ2
Therefore further limit can be simplified as follows,
limθ01cosθθcosθ=limθ02sin2(θ2)θcosθ
Multiplying and dividing by (θ2)2, we get
limθ02sin2(θ2)θcosθ×(θ2)2(θ2)2=limθ02θcosθ×θ24×sin2(θ2)(θ2)2
=limθ02cosθ×θ4×1
=2cos0×04
=0
Hence, value of limit is 0.
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