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(a) Use the Squeeze Theorem to evaluate $ \displaystyle \lim_{x \to \infty} \frac{\sin x}{x} $. (b) Graph $ f(x) = (\sin x)/x $. How many times does the graph cross the asymptote?

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Oregon State University

Harvey Mudd College

University of Nottingham

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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This is problem number sixty five of this tour Calculus eighth edition, Section two point six. Use the squeeze staring to evaluate the Limited's expertise in penalty. A sign of X over Now are squeezed there, Miss Cui recall is if we have two three functions age Afrin, G and F is a function that is greater than or equal to h but less than or equal to G and the limit as experts. Zane, some value ain of age is equal to the limit. Is expert is eight for the function gene, and this is they're both equal to each other to some limit at all. Therefore, for these reasons, the result is that the limit is expert is a for the function have it is also equal down and that is the definition of are squeezed here. Um, we're going do this for our problem here. Com. We know that sign of X is a function that arse awaits between the values negative or a positive one. If we divide each turn by X, we get negative. Wanna Rex less than equal to sign X or ex less than equal to one a rex. And now, if we just take the limit. Ah, Of each term as ex president, well has expressions impunity for negative One of rags approaches zero since a very large number of the denominator it means of this number increasingly grows smaller. So the limit hero be zero. The same reasoning goes to proving that the limit its export is infinity. One over hex goes through. So that must mean they're because the planet is expressive unity of negative Wanna wreck zero And the Limited Express in India won a rec zero. Then the limited Express infinity of sand over Cenex Rex must also be zero. All right, preppie graph vexes sine x rex. How many times is a graft crosses the asked me to So when it talks about the Assam took, I recall that we define horizontal ass and tits as the limit and six persons infinity of a certain function. So technically, here we have proven that why equal zero is a horizontal as him total dysfunction on. So, by looking at the ground, we identify the horizontal awesome totals the X axis and we see that this function crosses thie XX is an infinite amount of time. Many, many times are oscillating up and down over the X axis, and the function approaches zero as expressions infinity but isn't exactly equal to zero. So the answers and an infinite amount of times.

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