Find positive a, b for that function has limit My problem is to find a &#x2265;<!-- ≥ -->

delirija7z

delirija7z

Answered question

2022-07-07

Find positive a, b for that function has limit
My problem is to find a 0, b 0 for for which the following limit:
lim x + ( 5 1 x + sin ( a x 3 + 3 π 2 ) + ln ( 1 + e b x e b x 2 ) )
exists and equals 0.
I think that answer is a=0, b=0. But I do not know how to accurately prove that
lim x + cos ( a x 3 ) exists and equals to 1 only if a=0;
lim x + ln ( 1 + sinh ( b x ) ) exists and equals to 0 only if b=0.
Please help me to prove these two statements.

Answer & Explanation

Jayvion Tyler

Jayvion Tyler

Beginner2022-07-08Added 23 answers

It's easy to show that if a=b=0 then sin ( a x 3 + 3 π 2 ) = sin ( 3 π 2 ) = 1 and ln 1 + e b x e b x 2 = ln 1 = 0 are constants and that lim 5 1 x = 1
Thus the limit is 0 if a=b=0.
If a 0 then lim sin ( a x 3 + 3 π 2 ) does not exist as sin will take on all values of from -1 to 1 repeatedly. To show this formally, show that for any M there will exist an x>M so that a x 3 + 3 π 2 = 2 k π + π 2 for some integer k and a y>M as that a y 3 + 3 π 2 = 2 j p i + 3 π 2 for some integer j. so there will always by x,y>M so that sin ( a x 3 + 3 π 2 ) = 1 and sin ( a y 3 + 3 π 2 ) = 1
So if a 0 ; b = 0 there will be no limit.
If b>0 then e b x = ( e b ) x and e b x 0 so ln ( 1 + e b x e b x 2 )
So if b 0 there will be no limit.

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