MHB What is the simplified form of the limit as x approaches infinity?

AI Thread Summary
The limit as x approaches infinity for the expression \(\lim_{x\to +\infty}\sqrt{x^2+3}-x\) can be simplified by multiplying by the conjugate \(\frac{\sqrt{x^2+3} + x}{\sqrt{x^2+3} + x}\). This approach helps eliminate the indeterminate form. Alternatively, using the Binomial Theorem, the limit can be expressed as \(\lim_{x\to +\infty}\left(x + \frac{3}{2x} - x\right)\), which simplifies further. Ultimately, the limit evaluates to \(\frac{3}{2}\). Understanding these methods is crucial for solving similar limits effectively.
karush
Gold Member
MHB
Messages
3,240
Reaction score
5
\lim_{x\to +\infty}\sqrt{x^2+3}-x

sorry first of all how do you turn this string into latex
I don't see the icon tool on the editor

thnx
 
Last edited by a moderator:
Mathematics news on Phys.org
karush said:
\lim_{x\to +\infty}\sqrt{x^2+3}-x}

sorry first of all how do you turn this string into latex
I don't see the icon tool on the editor

thnx
$$\lim_{x \to +\infty} \sqrt{x^2+3}-x$$

Click on Reply With Quote to see the LaTex. Or better yet, there is a "How To Use LaTex on This Site" thread in the LaTeX Help section of the site.

Hint: Try multiplying by $$ \frac{\sqrt{x^2+3} + x}{\sqrt{x^2+3} + x} $$
 
Last edited:
karush & edit said:
\lim_{x\to +\infty}{(\sqrt{x^2 + 3} - x)}

Redo post:Alternative
Using the Binomial Theorem for (x^2 + 3)^{\frac{1}{2}},

write the limit as \displaystyle\lim_{x\to +\infty}{\bigg(x + \dfrac{3}{2x} + (all \ \ other \ \ terms \ \ with \ \ degree \ \ less \ \ than \ \ -1) \ - \ x \bigg)}
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Back
Top