x → ∞ lim ( x 4 + a x 3 + 3 x 2 + b x + 2 − x 4 + 2 x 3 − c x 2 + 3 x − d )
The limit above equals to 4 for constants a , b , c , d .
What is the value of the expression below?
a + c + ( b − 5 4 d ) ( a − c 1 0 )
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Wow that's long, I guess that's the only way to solve it. Good job!
We can also multiply above and below by 1/(x^2) ; take the 1/(x^2) in the numerator inside both the roots where it becomes 1/(x^4) ; then adjust the expression inside the roots to get a sort of "inverse polynomial" just like Pi Han Goh's 3rd step's denominator; and then apply L'Hospital's as it is 0/0 form.
However it is essentially the same ...Or is it ??
I have solved same vay. But c and d constats confused me at the first moment! :)
I never checked for the required expression and went on the quest for finding b and d, which actually belong to R :P
using binomial expansion for rational index and neglecting higher degree terms one can easily evaluate this limit
Can you elaborate more?
You have mint expansion series at x = ∞ :
x 4 + a x 3 + 3 x 2 + b x + 2 − x 4 + 2 x 3 − c x 2 + 3 x + d =
= 2 1 ( a − 2 ) x + ( − 8 a 2 + 2 c + 2 ) + 1 6 x a 3 − 1 2 a + 8 ( b − c − 4 ) + … ?
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We multiply the expression in the limit by its "conjugate":
x 4 + a x 3 + 3 x 2 + b x + 2 + x 4 + 2 x 3 − c x 2 + 3 x − d x 4 + a x 3 + 3 x 2 + b x + 2 + x 4 + 2 x 3 − c x 2 + 3 x − d
The limit becomes:
= x → ∞ lim x 4 + a x 3 + 3 x 2 + b x + 2 + x 4 + 2 x 3 − c x 2 + 3 x − d ( x 4 + a x 3 + 3 x 2 + b x + 2 ) − ( x 4 + 2 x 3 − c x 2 + 3 x − d ) x → ∞ lim x 2 ( 1 + x a + x 2 3 + x 3 b + x 4 2 + 1 + x 2 − x 2 c + x 3 3 − x 4 d ) x 3 ( a − 2 ) + x 2 ( c + 3 ) + x ( b − 3 ) + ( d + 2 )
Focusing on the denominator, for x → ∞ , the denominator → 2 x 2
If a = 2 , the the limit becomes O ( x 2 ) O ( x 3 ) → ∞ , thus a = 2
Now the expression of the limit is in the of O ( x 2 ) O ( x 2 ) , with the ratio of coefficient of x 2 in the numerator and denominator as 2 c + 3 , which equals to the given value 4 , thus c = 5 , and so
a − c 1 0 = 2 − 5 1 0 = 0 ⇒ ( b − 5 4 d ) ( a − c 1 0 ) = 0
Hence, our answer is simply a + c = 2 + 5 = 7