Limits Problems (1 Viewer)

[locked.on]

Need a little help with solving these limits. Thanks in advance.​

 

[Trebla]

√[x(x+1)] - x
Multiply by {√[x(x+1)] + x} / {√[x(x+1)] + x}
= [x(x + 1) - x²] / {√[x(x+1)] + x}
= [x² + x - x²] / {√[x(x+1)] + x}
= x / x{√(x+1)] / √x + 1}
= 1 / {√[(x+1)] / √x + 1}
As x approaches infinity, √[(x+1)] / √x approaches 1
lim { √[x(x+1)] - x } = 1 / (1 + 1) = 1/2
x--> ∞

√[2x(x+1)] - x
Multiply by {√[2x(x+1)] + x} / {√[2x(x+1)] + x}
= [2x(x + 1) - x²] / {√[2x(x+1)] + x}
= [2x² + 2x - x²] / {√[2x(x+1)] + x}
= [x² + 2x] / {√[2x(x+1)] + x}
Multiply by (1/x²) / (1/x²)
= [1 + 2/x] / {√[2x(x+1)] / x² + 1/x}
= [1 + 2/x] / {√[2x(x+1)] / x² + 1/x}
= [1 + 2/x] / {√[2(x+1)] / x√x + 1/x}
= [1 + 2/x] / {(1/x)√[2(x+1)] /√x + 1/x}
As x approaches infinity, √[2(x+1)] /√x approaches √2 and 1/x approaches zero, so the total limit is undefined i.e.
lim { √[2x(x+1)] - x } is undefined
x--> ∞

Not too sure with these, but I think they're right...

 

[vafa]

View attachment 16930

 

[locked.on]

Thanks for the help. Much needed.