How to find the value of a and b from this limit problem with or without L'Hopital's formula?Consider the limit: lim x → 4 x 2 + a x + b x − 4 = 14Question: How can I find the values of a and b?Attempt:My first thought is, we need to use L'Hopital's rule to make sure that the denominator isn't zero:Applying L'Hopital's rule, and we get: lim x → 4 2 x + a 1 = 14Then, we can substitute the limit of x to the equation such that: 2 ( 4 ) + a = 8 + a = 14and we get that the value of a is 6.But, how can I find the value of b? It seems that after applying the L'Hopital's formula the value of b disappears.Also, is there a way to solve this problem without L'Hopital's rule?Thanks

Question

How to find the value of a and b from this limit problem with or without L&#039;Hopital&#039;s formula?Consider the limit:      lim          x      →      4                          x        2            +      a      x      +      b              x      −      4        =  14Question: How can I find the values of a and b?Attempt:My first thought is, we need to use L&#039;Hopital&#039;s rule to make sure that the denominator isn&#039;t zero:Applying L&#039;Hopital&#039;s rule, and we get:      lim          x      →      4                  2      x      +      a        1    =  14Then, we can substitute the limit of x to the equation such that:  2  (  4  )  +  a  =  8  +  a  =  14and we get that the value of a is 6.But, how can I find the value of b? It seems that after applying the L&#039;Hopital&#039;s formula the value of b disappears.Also, is there a way to solve this problem without L&#039;Hopital&#039;s rule?Thanks

enfeinadag0 · Accepted Answer

To solve this without L&#039;Hopital, observe that since the top of the fraction in the limit is a quadratic polynomial, we can rewrite the limit as      lim          x      →      4                  (      x      −      α      )      (      x      −      β      )              x      −      4        =  14where   α  ,  β  ∈      C   are the roots of the polynomial       x    2    +  a  x  +  bIn order for the limit to exist, as   x  →  4  since the bottom of the fraction tends to 0, we must have       x    2    +  a  x  +  b  →  0  so in particular, at least one of   α  ,  β must be 4. Say   α is 4 (swapping the order doesn&#039;t change anything).Then the limit becomes       lim          x      →      4        (  x  −  β  )  =  14 which can be easily solved to give   β  =  −  10Now equating the coefficients of       x    2    +  a  x  +  b  =  (  x  −  α  )  (  x  −  β  )  =  (  x  −  4  )  (  x  +  10  )gives the required values for a and b.

sweetymoeyz · Accepted Answer

You need that limit to have the   0      /    0 indeterminate form. So      4    2    +  4  a  +  b  =  0    ⟹    b  =  −  16  −  4  a  .But you found out that    a  =  6. So   b  =  −  40