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Question: \(\operatorname { Lim } _ { x \rightarrow - 1 }\) \(\frac{\cos 2 - \cos 2x}{x^{2} - |x|}\)equals...

Lim⁡x→−1\operatorname { Lim } _ { x \rightarrow - 1 } cos⁡2−cos⁡2xx2−∣x∣\frac{\cos 2 - \cos 2x}{x^{2} - |x|}equals

A

sin 2

B

2 sin 2

C

4 sin 2

D

0

Answer

2 sin 2

Explanation

Solution

Lim⁡x→−1\operatorname { Lim } _ { x \rightarrow - 1 } cos⁡2−cos⁡2xx2+x\frac { \cos 2 - \cos 2 x } { x ^ { 2 } + x }

=Lim⁡x→−1\operatorname { Lim } _ { x \rightarrow - 1 } 2sin⁡(1+x)sin⁡(x−1)x(x+1)\frac { 2 \sin ( 1 + x ) \sin ( x - 1 ) } { x ( x + 1 ) }

Lim⁡x→−1\operatorname { Lim } _ { x \rightarrow - 1 } 2sin⁡(1+x)(1+x)⋅sin⁡(x−1)x=2sin⁡(−2)(−1)\frac { 2 \sin ( 1 + x ) } { ( 1 + x ) } \cdot \frac { \sin ( x - 1 ) } { x } = \frac { 2 \sin ( - 2 ) } { ( - 1 ) }× 1

= 2 sin 2