How do I solve for a and b in the following equation: lim x → 0 ( sin ⁡ ( a x ) + b − 2 tan ⁡ ( b x ) ) = 3 I need to find a value for a and b that is a Real Number. How would I go about doing this? Can Mathmatica help solve this equation?

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How do I solve for a and b in the following equation:      lim          x      →      0            (                  sin        ⁡        (        a        x        )        +        b        −        2                    tan        ⁡        (        b        x        )              )    =  3 I need to find a value for a and b that is a Real Number. How would I go about doing this? Can Mathmatica help solve this equation?

Sydnee Villegas · Accepted Answer

Clearly we must have   b  ≠  0 otherwise denominator of the function (whose limit is given in the question) will be zero. Now we are given that      lim          x      →      0                  sin      ⁡      (      a      x      )      +      b      −      2              tan      ⁡      (      b      x      )        =  3Clearly we can see that      lim          x      →      0        sin  ⁡  (  a  x  )  +  b  −  2  =      lim          x      →      0                  sin      ⁡      (      a      x      )      +      b      −      2              tan      ⁡      (      b      x      )        ⋅  tan  ⁡  (  b  x  )  =  3  ⋅  0  =  0or 0+b−2=0 so that b=2. Next note that if a=0 then we have the numerator   sin  ⁡  (  a  x  )  =  0 identically and hence the limit will not be 3. Therefore   a  ≠  0We now have      lim          x      →      0                  sin      ⁡      (      a      x      )              tan      ⁡      (      b      x      )        =      lim          x      →      0                  sin      ⁡      (      a      x      )              a      x        ⋅      a    b    ⋅            b      x              tan      ⁡      (      b      x      )        =  1  ⋅      a    b    ⋅  1  =      a    2  Thus   a      /    2  =  3 and hence a=6. We thus have a=6,b=2.It is important to show that both a,b are non-zero in order to get the correct answer.

How do I solve for a and b in the following equation: <munder> <mo movablelimits="true" f

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