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Problem 27 Medium Difficulty

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \left(\sqrt{9x^2 + x} - 3x \right) $

Answer

$$
\lim _{x \rightarrow \infty}\left(\sqrt{9 x^{2}+x}-3 x\right)=\frac{1}{6}
$$

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Video Transcript

in this problem, we want to find the limit of this function F of X as X approaches positive infinity. So as we're moving uh in the positive direction. On the X axis, on and on and on forever towards positive infinity. Uh We're going to investigate this limit graphically. So we used Gizmos um and I typed into function F of X equals the square root of nine X squared plus X. Uh Subtracting three X. With the help of this keypad down here. If you take a look. Uh Now if we want to look at the function values, you want to look at the y coordinate how high the graph is reaching. If we look at this graph, you can see that it almost looks like it's holding constant. So as ex moves uh towards positive infinity as we keep moving to the right, you can see that this graph is pretty constant. What is the value of F of X? Uh Somewhere along this graph? Well, we'll just click on one of these points and look for the Y coordinate. That would be the value of effort. Becks here. You can see F of X is 0.166. So that would be 16. Double check that with the calculated real quick. But I believe One divided by six is .1666. It is so it looks like the function value Is um won over six or .166? If we want to move further to the right to see if the function uh maybe you're starting to climb a little bit. Let's investigate the function value the Y coordinate uh Down here. Uh still holding the same .167. Let's try a little bit more. Remember. We want to find the limit as X approaches positive infinity. We're investigating this limit graphically One last time. Let's find AY coordinate once again, .167. Uh so it's fair to say that the limit of F of X as X approaches positive infinity Is 1/6 or .167.