How to find the value of a and b from this limit problem with or...

DIAMMIBENVERMk1

DIAMMIBENVERMk1

Answered

2022-07-07

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 = 14
Question: 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 = 14
Then, we can substitute the limit of x to the equation such that:
2 ( 4 ) + a = 8 + a = 14
and 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

Answer & Explanation

enfeinadag0

enfeinadag0

Expert

2022-07-08Added 16 answers

To solve this without L'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 = 14
where α , β C are the roots of the polynomial x 2 + a x + b
In 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't change anything).
Then the limit becomes
lim x 4 ( x β ) = 14
which can be easily solved to give β = 10
Now 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

sweetymoeyz

Expert

2022-07-09Added 8 answers

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

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