limit as x-> infinity

[PaulKraemer]

Hi,

Can anyone help me find...

lim as x -> infinity of [ sqrt (x^2 + 5x) - x]

...in all of the examples in this chaper, the functions are always a quotient. You divide both the numerator and the denominator by the highest power of x and then it makes the limit easy to figure out. In this one where the function is not expressed as a quotient, I can't figure out how to get started.

Any help will be greatly appreciated. Thanks in advance.

Paul

 

[Deleted member 4993]

Hi,

Can anyone help me find...

lim as x -> infinity of [ sqrt (x^2 + 5x) - x]

...in all of the examples in this chaper, the functions are always a quotient. You divide both the numerator and the denominator by the highest power of x and then it makes the limit easy to figure out. In this one where the function is not expressed as a quotient, I can't figure out how to get started.

Any help will be greatly appreciated. Thanks in advance.

Paul

\(\displaystyle \sqrt{x^2-5x} \ - \ x \ \)

\(\displaystyle = \ (\sqrt{x^2-5x} \ - \ x) \cdot \frac{\sqrt{x^2-5x} \ + \ x }{\sqrt{x^2-5x} \ + \ x } \)

\(\displaystyle = \ \frac{-5x}{\sqrt{x^2-5x}\ + \ x} \ \)



\(\displaystyle = \ \frac{-5}{\sqrt{1-\frac{5}{x}}\ + \ 1}\)

Now continue......

 

[PaulKraemer]

Thank you Subhotosh,

That was a big help!

Kind regards,
Paul

 

[lookagain]

Hi,

Can anyone help me find...

lim as x -> infinity of [ sqrt (x^2 + 5x) - x]


Paul

PaulKraemer,

please go with the method of Subhotosh Khan, if you would like.


\(\displaystyle Let \ ** \ = \ [All \ remaining \ terms \ in \ the \ infinite\ series \ \ have\ degrees \ \le -1].\)


\(\displaystyle Also, \ using \ the \ Binomial \ Theorem,\)

\(\displaystyle \sqrt{x^2 \ - \ 5x} \ =\)

\(\displaystyle (x^2 \ - \ 5x)^{\frac{1}{2}} \ =\)

\(\displaystyle \ (x^2)^{\frac{1}{2}} \ + \ \frac{1}{2}(x^2)^{\frac{-1}{2}}(-5x)^1 \ + \ ** \ =\)

\(\displaystyle x \ - \ \frac{5}{2} + **\)


\(\displaystyle Then, \ take \ the \ limit \ (as \to \infty) \ of \ [(x \ - \ \frac{5}{2} \ + \ **) \ - \ x].\)