Get the answer to "Evaluate the following limits.limθ→0sec⁡θ−1θ"

Marzadri9lyy

Answered question

2022-03-28

Evaluate the following limits.

lim_{θ \to 0} \frac{\sec θ - 1}{θ}

cineworld93uowb

Beginner2022-03-29Added 16 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.

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