17–83. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_u→ π/4 (tan u - cot u) / (u - π/4)

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Calculate the value of the given limit. Apply Lahapatal's rule if necessary. The limit as X approaches pi divided by 4 of tangent of X minus 1, all divided by pi divided by 4 minus X. Awesome. So it appears for this particular problem what we're ultimately trying to solve for is we're trying to calculate the value of this given particular limit. So now that we know that we're trying to figure out what the value of this limit is, let's read off our multiple choice answers to see what our final answer might be. So A is 2, B is 1, C is -1, and D is -2. Fantastic. So first off, let us apply the direct substitution method to see if we will get an indeterminate form as an answer, because in order to apply Lajapatal's rule, if it's necessary, that's a hint given by given to us by the prom itself. So in order to use Laja Patal's rule, we have to determine if we use. Direct substitution if it gives us an indeterminate form. But what does that mean? What's an indeterminate form? Well, an indeterminate form is we get an answer of 0 divided by 0 or infinity divided by infinity. So let us plug in a value of, since once again we're told that x approaches pi divided by 4, let us plug in a value that x equals pi divided by 4 directly into our expression to C. What value we will get. So when we do that, we will find that tangent of pi divided by 4 minus 1 divided by pi divided by 4 minus pi divided by 4 means we have to plug that in for our value for x. It will give us an answer of 0 divided by 0 when we evaluate it in our calculator or when we plug this expression into our calculator. So because we have an indeterminate form of zero divided by 0, we can therefore use La Hait's rule. We can recall and use it since we receive an indeterminate form after we use the direct substitution method. So, for evaluating limits with indeterminate forms, once again, such as 0 divided by 0 or infinity divided by infinity, we can recall and apply Lahapa's rule to solve those types of limits. So as we should recall a note, we should note that Lahapital's rule states that for calculating limits with indeterminate forms, we can differentiate the numerator and denominator separately and repeat this particular process until we stop receiving an indeterminate form when we use the direct substitution method. So with that said, therefore we need to recall and apply Lahapital's rule to solve our limit that we're asked to evaluate. And to do that, we must first find the derivatives of the numerator and denominator. So when we differentiate the numerator, which is the top expression, and of course the denominator is the bottom expression in our division here, so when we differentiate the numerator with respect to X, we could say that D by DX, where once again D by DX is just a fancy way of saying the derivative with respect to X. So if we take the take D by DX of tangent of X minus 1. We will find that the derivative is equal to C2X, and if we differentiate the denominator with respect to X, we will get that D by DX of pi divided by 4 minus X is going to be equal to 0 minus 1, which is going to give us an answer of -1, fantastic. So now we can apply Lahapa's rule, and as we should recall a note, we can go ahead and write that the limit as X approaches pi divided by 4 of the tangent of X minus 1, all divided by divided by 4 minus X is equal to the limit, as X approaches pi divided by 4 of. C can squared of X divided by -1. Which will give us, which we can simplify to negative c can of pi divided by 4. Once again, we are, we can pull out the -1 to the front. Since anything divided by 1 is going to be, so we can bring out the negative, and we can state that so now when we do direct substitution, since we do a little bit of simplifying and we do direct substitution, since we now know, let's make a note here that our X equals pi divided by 4. So if we plug in pi divided by 4 in for X, we will find that negative C can't. Of pi divided by 4 to the power 2 is going to be equal to negative square rooted 22, which when we do plug that into our calculator, we will get a value of -2. As our final answer, when we plug that value of the square rooted 2 squared or don't forget your negative, so negative square rooted 2 squared into our calculator, we will get an answer of -2. So therefore, without a doubt, our final answer has to be multiple choice answer letter D, which once again is -2. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_u→ π/4 (tan u - cot u) / (u - π/4)

Key Concepts

Limits

L'Hôpital's Rule

Trigonometric Functions

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