Q. 97

Question

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 that $f (x) = c o s x$ is continuous everywhere.

Step-by-Step Solution

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Answer

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

1Step 1. Given Information:

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

2Step 2. Prove:

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

Q. 96

Q. 98

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