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Problem 30 Easy Difficulty

(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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Video Transcript

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