a. Estimate lim x→π/4 cos 2x / cos x − sin x by making a table...

Question

a. Estimate lim x→π/4 cos 2x / cos x − sin x by making a table of values of cos 2x / cos x − sin x for values of x approaching π/4. Round your estimate to four digits.

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with limits. This problem says to determine the given limit with the help of a table, and we're looking at the limit as X approaches 3 pi divided by 4 of the function of cosine of 2 X divided by the quantity of sin X plus cosine x. We have 4 possible choices as our answers. For choice A, we have 0. for choice B, we have minus the square root of 2. For choice C, we have the square root of 2, and for choice D, we have D and E, which represents does not exist. Now, we're asked to use a table to determine this limit. And so, the X value that we're looking for to find the limit of is 3 pi. Divided by 4. We actually need this as a decimal, so this is going to be approximately equal to. 2.356, and here we've just rounded to three decimal places. So, we're going to make it a table with two rows. The first row is going to be labeled X, and the second row is going to be F of X. And we're going to look at several different values of X. So, the first value that we're going to look at is 2.2. The next value is going to be 2.25. The next value is going to be 2.3. The next value is going to be 2.4. The next value is going to be 2.45. And then finally, our last value is going to be 2.5. So, we have a total of 6 values that we need um to plug in for X and calculate the value of our function. So, we're gonna do that. So, we're gonna start off with X equal to 2.2. So, we're gonna be looking at F of 2.2. And so here we're just going to be using the function that we have to take the limit of. So F of 2.2 is going to be equal to the cosine of the quantity of 2 multiplied by 2.2 in quantity, divided by the quantity of sine of 2.2 plus cosine of 2.2. So using our calculator to evaluate this expression, this is going to be approximately equal to minus 1.397. And here we just kept three decimal places. So we'll add that in the first blank column. In our tables, so we'll have -1.397 in that spot. So, now we're just going to repeat this calculation for the other X values. So we'll have the F of 2.25. This is going to be equal to the cosine of the quantity of 2 multiplied by 2.25. In quantity Divided by the quantity of the sine of 2.25 plus cosine of 2.25. Again, we can use our calculators to evaluate this expression. And we will get -1.407. So we will add that value. Into the next slot in our table. So moving on to the next X value, which is 2.3, so we'll have F of 2.3. And this is going to be equal to the cosine of the quantity of 2 multiplied by 2.3. In quantity That's gonna be divided by the quantity of the sine of 2.3 plus cosine of 2.3. And again, using our calculator to evaluate this expression, this will be approximately equal to -1.412. So, we'll add that value. To the next column. So, the next X value that we need to evaluate is 2.4, so we'll have F of 2.4. And this is going to be equal to the cosine of the quantity of 2 multiplied by 2.4 in quantity. That's gonna be divided by the quantity of sine. Of 2.4 plus cosine of 2.4. And again, using our calculators, this is gonna be approximately equal to minus 1.413. We'll add that to the next column. In our table. So the next value we need to um look at is X is equal to 2.45. So we'll have F of 2.45. This is going to be equal to cosine of the quantity of 2 multiplied by 2.45. In quantity, that's gonna be divided by the quantity of sign. Of 2.45 plus cosine of 2.45. And again, using our calculators, this will be approximately. -1.408. So we'll add that value into the next column. And then our final value that we need to look at is X equal to 2.5. So we'll have F of 2.5. This is going to be equal to the cosine of the quantity of 2 multiplied by 2.5 in quantity. Divided by the quantity of sign. Of 2.5 plus cosine of 2.5. And this is going to be approximately equal to minus. 1.400. And so we'll add that value into the last column in our table. So now that we have all the values on our table, we can use it to determine the limit that we're given in this problem. And so the limit as X approaches 3 pi divided by 4, that actually falls in between 2.3 and 2.4. And notice that as we approach. 2.3, that our value is actually decreasing, it's getting more and more negative. But then afterwards, as from 2.4 onwards, the value is actually increasing, it's becoming less negative. So there must be a minimum in between 2.3 and 2.4. So its value is probably going to be less than both of those values. So if we look at the limit, As X approaches. 3 pi divided by 4. Of the function of cosine of 2 X. Divided by the quantity of sin x. Plus cosine X. Using this table, this is probably going to be approximately equal to minus 1.414, because it's going to be slightly less than both of those values. And so, for the value of X is equal to 2.4, we had minus 1.413, and so we'll just go one more decimal place, or one more um 1000th decimal place below that. So we'll get -1.414. Now, minus 1.414 is equal to minus the square root of 2, if we were to round that value to three decimal places. So, this is actually the value that we're looking for. And if we come over to our answer choices, this actually corresponds to answer choice B. So, I hope that this explanation has been useful, and I'll see you in the next video.

a. Estimate lim x→π/4 cos 2x / cos x − sin x by making a table of values of cos 2x / cos x − sin x for values of x approaching π/4. Round your estimate to four digits.

Key Concepts

Limits

Trigonometric Functions

Table of Values

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