Chapter 1: Q. 23 (page 135)

Use a geometric argument and the Squeeze Theorem for limits to argue that $lim_{θ \to 0^{-}} \cos θ = 1$
for sufficiently small negative angles θ.

Short Answer

Expert verified

$lim_{θ \to 0^{-}} \cos θ = 1$ for sufficiently small negative angles θ has been proven.

Step by step solution

Step 1. Given information.

We have to use a geometric argument and the Squeeze Theorem for limits to argue that $lim_{θ \to 0^{-}} \cos θ = 1$

Step 2. Prove the argument

Look at the picture below :

Using Squeeze Theorem to calculate limits of $\cos θ$ at x=0, take angle measured in radians and $- \frac{π}{4} < θ < 0$

According to the figure, we clearly have $0 < 1 - \cos θ < θ$

Thus by Squeeze Theorem, we have,

$\begin{aligned} lim_{θ \to 0^{-}} (1 - \cos θ) = lim_{θ \to 0^{-}} 1 - lim_{θ \to 0^{-}} \cos θ \\ = 1 - \cos 0 \\ = 1 - 1 \\ = 0 \end{aligned}$

Step 3. Rewrite the limits

Rewrite the limits:

$\begin{array}{r} lim_{θ \to 0^{-}} (1 - \cos θ) = 0 \\ lim_{θ \to 0^{-}} 1 - lim_{θ \to 0^{-}} \cos θ = 0 \\ 1 - lim_{θ \to 0^{-}} \cos θ = 0 \\ lim_{θ \to 0^{-}} \cos θ = 1 \end{array}$

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Use a geometric argument and the Squeeze Theorem for limits to argue that $lim_{θ \to 0^{-}} \cos θ = 1$
for sufficiently small negative angles θ.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Prove the argument

Step 3. Rewrite the limits

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Use a geometric argument and the Squeeze Theorem for limits to argue that limθ→0− cos⁡θ=1for sufficiently small negative angles θ.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Prove the argument

Step 3. Rewrite the limits

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Use a geometric argument and the Squeeze Theorem for limits to argue that $lim_{θ \to 0^{-}} \cos θ = 1$
for sufficiently small negative angles θ.