What mistake have I committed in doing these steps to find the limit: lim x → 0 cos ⁡ x − 1 x 2 ?

Question

What mistake have I committed in doing these steps to find the limit:       lim          x      →      0                  cos      ⁡      x      −      1              x      2       ?

Sarahi Thomas · Accepted Answer

lim          x      →      0                  cos      ⁡      x      −      1              x      2        =  −      lim          x      →      0                  2              sin        2            ⁡              x        2                    x      2        =  −      1    2        lim          x      →      0                  (                        sin          ⁡                      x            2                                    x          2                    )        2    =  −      1    2              (              lim                  x          →          0                                      sin          ⁡                      x            2                                    x          2                    )        2    =  −      1    2  For clarity, we may put   x  =  2  y  ,  x  →  0    ⟹    y  =      x    2    →  0  =  −      1    2              (              lim                  x          →          0                                      sin          ⁡                      x            2                                    x          2                    )        2    =  −      1    2              (              lim                  y          →          0                                      sin          ⁡          y                y            )        2    =  −      1    2

What mistake have I committed in doing these steps to find the limit: lim_(x => 0) (cos x-1)/(x^2) ?

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New Questions in Calculus 1