For what values of a and b is the following equation true? lim x →...

Karina Trujillo

Karina Trujillo

Answered

2022-06-26

For what values of a and b is the following equation true?
lim x 0 ( sin ( 2 x ) x 3 + a + b x 2 ) = 0

Answer & Explanation

knolsaadme

knolsaadme

Expert

2022-06-27Added 16 answers

We have
L = lim x 0 sin ( 2 x ) x 3 + a + b x 2 = lim x 0 sin ( 2 x ) + a x 3 + b x x 3
which is a limit of type 0 / 0, so we try L'Hôpital's rule:
L = lim x 0 2 cos ( 2 x ) + 3 a x 2 + b 3 x 2
The denominator again vanishes for x 0 , the nominator goes to 2 + b. So if b 2, the nominator does not vanish and we have
L = sgn ( 2 + b )
For b = 2 we again have a limit of type 0 / 0 and apply the rule again:
L = lim x 0 4 sin ( 2 x ) + 6 a x 6 x
The nominator and denominator vanish and we apply the rule once again:
L = lim x 0 8 cos ( 2 x ) + 6 a 6 = 6 a 8 6 = a 4 3
This gives the answer that the equation is true, L vanishes, if a = 4 / 3 and b = 2

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