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Compute the indicated limit.$$\lim _{x \rightarrow \infty} \frac{x}{1+x^{2}}$$

$$0$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

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Lectures

04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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04:01

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So we have those limits as except religious Infinity. Right, Uh, one minus X squared over. Um x que right. Minus x plus one minus X plus one. What is the highest power of X? That is three. Right. So we divide each and every tournament by the highest power of X. So one over X cubed, minus X squared over X cube, right minus X Q over X cube. Right minus X over X Q Right. Plus one over X cubed. Right. This is how you do. You do limits involving infinities. Right. Then this one cancels 12 of these. They're gonna have one here. This one cancels this one that's going cancels. One of these are gonna have to here. So, actually, what is gonna be left is one over X cube, then minus one over X right over one, right minus one over X squared, plus one over X cube. That is what you have left. And you find it a limit as X approaches. Infinity of that now asked, Eggs approaches infinity. You can see that this one will eventually go to zero. This one will also eventually go to zero. Because you have accepted denominator. Right. So the numerator is gonna be zero, right? And then this one, the denominator here is gonna eventually go to Syria. Here is gonna eventually go to zero. So I just have one, right? Just one is what is gonna be left. But 01 is still zero. So the answer is gonna be zero. So watch how we did it. That is how we're gonna do it. Uh uh.

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