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(a) Estimate the value of $$ \lim_{x \to 0}\frac{\sin x}{\sin \pi x} $$by graphing the function $ f(x) = (\sin x)/(\sin \pi x) $. State your answer correct to two decimal places.

(b) Check your answer in part (a) by evaluating $ f(x) $ for values of $ x $ that approach 0.

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 2

The Limit of a Function

Limits

Derivatives

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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 thirty of Stuart Calculus, eighth edition, Section two point two. Party has to meet. The value of the limit is experts zero of sign of X threated by sign. Ah, pi X, I graphing the function of state Your answer correct to two decimal, please. So we are going to take this function sign of X divided by sign of pranks. And we're going to plant this function around this area. X goes to zero. You do this with your graphing calculator or any other graphing device here in our spreadsheet. We have set up this plot of this function and we assumed an already very close to zero. And now we're preserving exactly where this my cross, the y axis. You see that it approaches from the left in a hurry. That purchase value approximately zero point three two. But it may be more Kurt to say, a value, lest then zero point three two. However, using a graph is not the most direct answer. So for the time being for party, we're going to estimate this. Yes, we can and say, that is, um it is approximately zero plane three two for part B we're going to check her answer and evaluate the function for various values that are close to zero. And in this way we may be able to estimate the limit further in a spreadsheet. We have used some values to plot the function here, and we're gonna take a look at this list of values and determined what the limit might be. We have values here that approach zero from the right, and we see that the values decrease closer and closer to a number. Their point three one eight three Ah, and so on. Same thing from the left hand side, we get about the same values. All the calculations are mirrored about to this y axis. And finally, what we can conclude is that the actual limited is closer to about zero point three one eight, which is a very close to our estimation of zero point three two term party. We're going to say that the limit is actually approximately zero point three one eight, which is a slight Lee better estimate than party

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