"Evaluate the following limits. limθ→0sec⁡θ−1θ" was answered by our community

cistG

Answered question

2020-11-23

Evaluate the following limits.

lim_{θ \to 0} \frac{\sec θ - 1}{θ}

Tuthornt

Skilled2020-11-24Added 107 answers

We have to find the limit:

lim_{θ \to 0} \frac{\sec θ - 1}{θ}

We know the identity,

\cos θ \sec θ = 1

\Rightarrow \sec θ = \frac{1}{\cos θ}

Substituting above value in the limit,

lim_{θ \to 0} \frac{\sec θ - 1}{θ} = \frac{\frac{1}{\cos θ} - 1}{θ}

= lim_{θ \to 0} \frac{\frac{1 - \cos θ}{\cos θ}}{θ}

= lim_{θ \to 0} \frac{1 - \cos θ}{θ \cos θ}

We have identity,

\cos^{2} x - \sin^{2} x = \cos 2 x

\Rightarrow 1 - \sin^{2} x - \sin^{2} x = \cos 2 x

1 - \cos 2 x = 2 \sin^{2} \frac{θ}{2}

then for

1 - \cos 2 x = 2 \sin^{2} \frac{θ}{2}

Therefore further limit can be simplified as follows,

lim_{θ \to 0} \frac{1 - \cos θ}{θ \cos θ} = lim_{θ \to 0} \frac{2 \sin^{2} (\frac{θ}{2})}{θ \cos θ}

Multiplying and dividing by

(\frac{θ}{2})^{2}

, we get

lim_{θ \to 0} \frac{2 \sin^{2} (\frac{θ}{2})}{θ \cos θ} \times \frac{(\frac{θ}{2})^{2}}{(\frac{θ}{2})^{2}} = lim_{θ \to 0} \frac{2}{θ \cos θ} \times \frac{θ^{2}}{4} \times \frac{\sin^{2} (\frac{θ}{2})}{(\frac{θ}{2})^{2}}

= lim_{θ \to 0} \frac{2}{\cos θ} \times \frac{θ}{4} \times 1

= \frac{2}{\cos 0} \times \frac{0}{4}

= 0

Hence, value of limit is 0.

Jeffrey Jordon

Expert2022-08-24Added 2605 answers

Answer is given below (on video)

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