I have to calculate the following limit lim x → + ∞ x ln ⁡...

Maliyah Robles

Maliyah Robles

Answered

2022-07-04

I have to calculate the following limit
lim x + x ln x 1 sin x

Answer & Explanation

Dalton Lester

Dalton Lester

Expert

2022-07-05Added 12 answers

lim n n 2 + ± ( 1 ) n
is infinity even though the denominator does not converge.
The real problem with your function is that it isn’t defined for all of R . But if f is a function defined on a subset of R with no real upper bound, we can still define lim x f ( x ) .
In your case, you easily get, where f is defined, and x > 1 ,
f ( x ) = x ln x 1 sin x x ln x 2 ,
since 0 < 1 sin x 2 in the domain of f
Since x ln x 2 , this means f ( x ) .
racodelitusmn

racodelitusmn

Expert

2022-07-06Added 5 answers

The quotient x ln ( x ) 1 sin ( x ) is undefined when x π 2 + 2 π Z . Otherwise, 1 sin ( x ) 2, and therefore
(1) x ln ( x ) 1 sin ( x ) x ln ( x ) 2 .
Since
lim x x ln ( x ) = ,
it follows from (1) that
lim x x ln ( x ) 1 sin ( x ) =
too.

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