Is the Limit of sqrt(x+sinx)ln(x) as x Approaches Zero Solvable?

[mathgeek69]

1. lim as x->o+ ( (sqrt(x+sinx))(lnx))



2. Homework Equations : [PLAIN]http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/lHopital/Laws/multiplication_law.gif[/b]



3. The Attempt at a Solution :

lim as x->o+ ( (sqrt(x+sinx))(lnx))
= lim as x->o+ (sqrt(x+sinx)) x lim as x->o+(lnx)
= 0(-∞)
= 0

 

[haruspex]

= 0(-∞)
= 0

0 × ∞ is indeterminate. When you get that result in a limits question it tells you you need to use a subtler approach.
What does sin(x) approximate as x tends to zero? Does that allow you to simplify it a little?
Can you see a variable substitution that would then allow you to lose the sqrt?

 

[mathgeek69]

as sin(x) approaches 0 as x approaches 0.

That didn't help me as I still resolve to 0*(-∞) but as you say we need a different approach.

I did try to do" let y = (x+sin(x))^2 " and try to take that approach but I didnt know what to do after as now I am introducing y all of a sudden and don't know how ln x is affected.

 

[haruspex]

as sin(x) approaches 0 as x approaches 0.

No, I was asking for some simpler function that it approximates on its way to 0.

 

[mathgeek69]

No, I was asking for some simpler function that it approximates on its way to 0.

It approximates cos(x+ (pi/2)) ?

 

[haruspex]

It approximates cos(x+ (pi/2)) ?

No, I said simpler. Here's a clue, do you know what the limit of sin(x)/x is as x tends to 0?

 

[mathgeek69]

no, i said simpler. Here's a clue, do you know what the limit of sin(x)/x is as x tends to 0?

0/0 ?

 

[HallsofIvy]

Then you need to review (or learn) limits.

 

[haruspex]

0/0 ?

0/0 is useless for the same reason that 0 × ∞ is useless. In these situation you get nowhere by taking the limit of each part independently.
What is the tangent to sin(x) at x = 0?