Math  /  Calculus

QuestionTrue or False \quad NO 10 x=0x=0y=sin1xy=\sin \frac{1}{x} 的振荡间 断点()  2. x=π/2 为 y=tanx 的无穷间断 \text { 2. } x=\pi / 2 \text { 为 } y=\tan x \text { 的无穷间断 }

Studdy Solution
Analyze the point x=π2 x = \frac{\pi}{2} for the function y=tanx y = \tan x :
The function y=tanx y = \tan x is defined as sinxcosx \frac{\sin x}{\cos x} . At x=π2 x = \frac{\pi}{2} , cosx=0 \cos x = 0 , which makes the function undefined. As x x approaches π2 \frac{\pi}{2} from the left, tanx \tan x approaches + +\infty , and from the right, it approaches -\infty . This is characteristic of an infinite discontinuity.
The statements are:
1. False: x=0 x = 0 is indeed an oscillating discontinuity for y=sin1x y = \sin \frac{1}{x} .
2. True: x=π2 x = \frac{\pi}{2} is an infinite discontinuity for y=tanx y = \tan x .

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