Chapter 1: Q. 97 (page 137)

In the reading, we used the Squeeze Theorem to prove that $\underset{h \to 0}{l i m} s i n h = 0$ and $\underset{h \to 0}{l i m} c o s h = 1$ . Use these facts, the sum identity for cosine, and limit rules to prove thatrole="math" localid="1648152969592" $f (x) = c o s x$ is continuous everywhere.

Short Answer

Expert verified

Ans: $\lim_{x \to c} c o s x = f (c)$

Step by step solution

Step 1. Given Information:

The strategy is to prove using Squeeze theorem, that $f (x) = c o s x$ is continuous everywhere.

Step 2. Prove:

$G i v e n c \in ℝ, \underset{x \to c}{l i m} f (x) = \underset{x \to c}{l i m} c o s x = \underset{h \to 0}{l i m} c o s (c + h) = \underset{h \to 0}{l i m} (c o s (c) c o s (h) - s i n (c) s i n (h)) = c o s (c) (\underset{h \to 0}{l i m} c o s h) - s i n (c) (\underset{h \to 0}{l i m} s i n h) = c o s c (1) - s i n c (0) = c o s c - 0 = c o s c = f (c)$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In the reading, we used the Squeeze Theorem to prove that $\underset{h \to 0}{l i m} s i n h = 0$ and $\underset{h \to 0}{l i m} c o s h = 1$ . Use these facts, the sum identity for cosine, and limit rules to prove thatrole="math" localid="1648152969592" $f (x) = c o s x$ is continuous everywhere.

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Prove:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Mechanics Maths

Geometry

Decision Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In the reading, we used the Squeeze Theorem to prove that limh→0sinh=0and limh→0cosh=1. Use these facts, the sum identity for cosine, and limit rules to prove thatrole="math" localid="1648152969592" f(x)=cosx is continuous everywhere.

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Prove:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Mechanics Maths

Geometry

Decision Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In the reading, we used the Squeeze Theorem to prove that $\underset{h \to 0}{l i m} s i n h = 0$ and $\underset{h \to 0}{l i m} c o s h = 1$ . Use these facts, the sum identity for cosine, and limit rules to prove thatrole="math" localid="1648152969592" $f (x) = c o s x$ is continuous everywhere.