Solve this : Q Check continuity for the following sin cos x1+ cos x - Maths - Continuity and Differentiability

Question

Solve this :

Q   Check   continuity   for   the  
following   sin   cos   x 1 +   cos   x
                 
                 
    x   ≠ 0          
        0          
                 
                 
                  x =
0 Where     denotes   Greatest   Integer  
Function

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

fx=sincosx1+cosxx≠00x=0We have f0=0NowL H L
=limx→0- fx=limx→0-sincosx1+cosxNow cos0=1 but as we move left of x=0, then value of cosx becomesless than 1 which means cosx=0 As X=0 if 0≤X<1⇒limx→0-sincosx1+cosx=sin01+0=0R
H L
=limx→0+ fx=limx→0+sincosx1+cosxNow cos0=1 but as we move right of x=0, then value of cosx becomesless than 1 which means cosx=0 As X=0 if 0≤X<1⇒limx→0+sincosx1+cosx=sin01+0=0Hencelimx→0- fx=limx→0+ fx=f0=0Hence function is continuous at x=0

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Solve this : Q . Check continuity for the following sin cos x 1 + cos x x ≠ 0 0 x = 0 Where denotes Greatest Integer Function

Solve this :

$Q . Check continuity for the following \{\begin{cases} \frac{\sin ([\cos x])}{1 + [\cos x]} x \neq 0 \\ 0 x = 0 \end{cases} Where [] denotes Greatest Integer Function$