Compute trigonometric limit without use of de L'Hospital's rule lim x → 0 ( x + c ) sin ⁡ ( x 2 ) 1 − cos ⁡ ( x ) , c ∈ R +

Question

Compute trigonometric limit without use of de L&#039;Hospital&#039;s rule      lim          x      →      0                  (      x      +      c      )      sin      ⁡      (              x        2            )              1      −      cos      ⁡      (      x      )        ,  c  ∈            R      +

Zackery Harvey · Accepted Answer

We can solve in two steps fistly multipling both sides to             x              2       and then to   1  +  cos  ⁡            (      x      )       and consider the fact       lim          x      →      0                          sin        ⁡        (                  x                      2                          )                    x                  2                      =    1   we will get      lim          x      →      0                  (      x      +      c      )      sin      ⁡      (              x                  2                    )              1      −      cos      ⁡      (      x      )        =      lim          x      →      0                  (      x      +      c      )                        x                          2                            (      1      −      cos      ⁡      (      x      )      )                  sin      ⁡      (              x                  2                    )                      x                    2              =    =      lim          x      →      0                  (      x      +      c      )                        x                          2                            (      1      −      cos      ⁡      (      x      )      )                  (      1      +      cos      ⁡      (      x      )      )              (      1      +      cos      ⁡      (      x      )      )        =      lim          x      →      0                  (      x      +      c      )                        x                          2                            1      −              cos                  2                    ⁡                        (          x          )                          (    1    +    cos    ⁡                  (        x        )              )      =      lim          x      →      0                  (      x      +      c      )                        x                          2                                    sin                  2                    ⁡                        (          x          )                          (    1    +    cos    ⁡                  (        x        )              )    =      lim          x      →      0            (    x    +    c    )        (    1    +    cos    ⁡                  (        x        )              )    =  2  c

Compute trigonometric limit without use of de L'Hospital's rule <munder> <mo movablelimits

Answered question

Answer & Explanation

New Questions in Differential Calculus