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Normally if we say a limit $L$ exists, we mean that $L$ is a finite number In what follows, we allow $L$ to stand for either $+\infty$ or $-\infty$. Let $$\lim _{x \rightarrow \infty} \frac{a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}}{b_{k} x^{k}+b_{k-1} x^{k-1}+\cdots+b_{0}}=L \text { If the integer } n-k>0, \text { show }$$ that $L$ is as given in Table 2 for each of the eight cases indicated:

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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.

So normally if we say that the limit exists, then we mean that L is a finite number. Um But what we can see is that L could also be a positive infinity or negative infinity. And what we see is that um if n is n minus K is greater than zero, so that means that N is greater than K. We want to show that the limit could be infinity or negative infinity. So let's simplify this and consider this rational function Acts to the 5th, divided by X cubed. We see in this case um that the limit as X goes to infinity is going to be a positive infinity and as it goes to negative infinity it'll be a positive infinity. That's because and is greater than K. If we make this perhaps a negative though, we see it will change around or if we add other components and it will change. But the general idea is we look at the degree of the highest degree term.

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