Chapter 1: Q. 98 (page 137)

Use the quotient rule for limits and the continuity of $\sin x$ and $\cos x$ to prove that $f (x) = \tan x$ is continuous on its domain.

Short Answer

Expert verified

It is proved that $f (x) = \tan x$ is continuous on its domain.

Step by step solution

Step 1. Given Information

We are given that $\sin x$ androle="math" localid="1648221237056" $\cos x$ are continous.

Step 2. Proving the statement

Given $c \in ℝ$ ,

$\lim_{x \to c} f (x) = \lim_{x \to c} \tan x = \lim_{h \to 0} \tan (c + h) = \lim_{h \to 0} \frac{\sin (c + h)}{\cos (c + h)} = \lim_{h \to 0} (\frac{\sin c \cos h + \cos c \sin h}{\cos c \cos h - \sin c \sin h}) = \frac{\sin c (\lim_{h \to 0} \cos h) + \cos c (\lim_{h \to 0} \sin h)}{\cos c (\lim_{h \to 0} \cos h) - \sin c (\lim_{h \to 0} \sin h)} = \frac{\sin c (1) + \cos c (0)}{\cos c (1) - \sin c (0)} = \frac{\sin c}{\cos c} = \tan c = f (c)$

Hence $\tan x$ is continous.

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Use the quotient rule for limits and the continuity of $\sin x$ and $\cos x$ to prove that $f (x) = \tan x$ is continuous on its domain.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

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Use the quotient rule for limits and the continuity of sinxand cosxto prove that f(x)=tanxis continuous on its domain.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

One App. One Place for Learning.

Most popular questions from this chapter

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Use the quotient rule for limits and the continuity of $\sin x$ and $\cos x$ to prove that $f (x) = \tan x$ is continuous on its domain.