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Find the limit or show that it does not exist.
$ \displaystyle \lim_{x \to \infty} \left(\sqrt{9x^2 + x} - 3x \right) $
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03:20
Daniel Jaimes
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 6
Limits at Infinity: Horizontal Asymptotes
Limits
Derivatives
Missouri State University
University of Michigan - Ann Arbor
Idaho State University
Lectures
04:40
In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
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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.
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