Question

$\lim _ { x \rightarrow 1 }$ $\frac{\sqrt{1 - \cos 2(x - 1)}}{x - 1}$

• A. Exists and it equal to$\sqrt{2}$
• B. Exists and it equals –$\sqrt{2}$
• C. Does not exist because x – 1 → 0
• D. Does not exist because the left hand limit is not equal to the right hand limit

Answer 1

Answer: (D)

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

Answer 2

$\lim _ { x \rightarrow 1 }$ $\frac { \sqrt { 1 - \cos 2 ( x - 1 ) } } { x - 1 }$ = $\lim _ { x \rightarrow 1 }$ $\frac { \sqrt { 2 } | \sin ( x - 1 ) | } { x - 1 }$

But $\frac { \sqrt { 2 } | \sin ( x - 1 ) | } { x - 1 }$

= $\sqrt { 2 }$ = $\sqrt { 2 }$ and

$\frac { \sqrt { 2 } | \sin ( x - 1 ) | } { x - 1 }$

= – $\sqrt { 2 }$ $\lim _ { x \rightarrow 1 - }$ = – $\sqrt { 2 }$ .