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Question: \[\lim_{x \rightarrow 1}\frac{\sqrt{1 - \cos 2(x - 1)}}{x - 1}\]...

lim⁡x→11−cos⁡2(x−1)x−1\lim_{x \rightarrow 1}\frac{\sqrt{1 - \cos 2(x - 1)}}{x - 1}

A

Exists and it equal 2\sqrt{2}

B

Exists and it equals −2- \sqrt{2}

C

Does not exist because x−1→0x - 1 \rightarrow 0

D

Does not exist because left hand limit is not equal to right hand limit

Answer

Does not exist because left hand limit is not equal to right hand limit

Explanation

Solution

f(1+)=lim⁡h→0f(1+h)==lim⁡h→01−cos⁡2hhf(1 + ) = \lim_{h \rightarrow 0}f(1 + h) = = \lim_{h \rightarrow 0}\frac{\sqrt{1 - \cos 2h}}{h}

=lim⁡h→02sin⁡hh2= \lim_{h \rightarrow 0}\sqrt{2}\frac{\sin h}{h\sqrt{2}}

f(1−)=lim⁡h→0f(1−h)=lim⁡h→01−cos⁡(−2h)−h=lim⁡h→02sin⁡h−h2f(1 - ) = \lim_{h \rightarrow 0}f(1 - h) = \lim_{h \rightarrow 0}\frac{\sqrt{1 - \cos( - 2h)}}{- h} = \lim_{h \rightarrow 0}\sqrt{2}\frac{\sin h}{- h\sqrt{2}}

∴\therefore limit does not exist because left hand limit is not equal to right hand limit.