How do you find the limit of x tan (9/x) as x approaches infinity using l'hospital's rule?

cearexceptotob3q

cearexceptotob3q

Answered question

2023-02-14

How to find the limit of x tan ( 9 x ) as x approaches infinity using l'hospital's rule?

Answer & Explanation

Vincent Haley

Vincent Haley

Beginner2023-02-15Added 3 answers

lim x x tan ( 9 x ) has indeterminate form 0 .
In order to use l'Hospital's rule we rewrite the expression:
x tan ( 9 x ) = tan ( 9 x ) 1 x
The limit is the rewritten expression has form 0 0 , so we can apply l"Hospital.
By separating the numerator and denominator, we get:
lim x x tan ( 9 x ) = lim x sec 2 ( 9 x ) ( - 9 x 2 ) - 1 x 2
= lim x ( sec 2 ( 9 x ) ( 9 ) )
= 9 sec 2 ( 0 ) = 9
We do not need l'Hospital
x tan ( 9 x ) = 9 sin ( 9 x ) 9 x 1 cos ( 9 x )
lim u sin u u = 1 , then we get
lim x x tan ( 9 x ) = lim x 9 sin ( 9 x ) 9 x 1 cos ( 9 x )
= ( 9 ) ( 1 ) ( 1 ) = 9

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