Problem

Evaluate the following limits or state that they …

08:48

Problem 67 Hard Difficulty

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)
$$\lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4}$$

Answer

2

Related Courses

Calculus 1 / AB

Calculus Early Transcendentals

Chapter 3

Derivatives

Section 5

Derivatives of Trigonometric Functions

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Video Transcript

So this question is essentially asking us to find the limit or the derivative of this tangent X minus one over X minus pi over four. And on the right hand side, I wrote essentially a form that will hopefully make this a little easier. Understand? So this off of X is essentially telling you the function. So the tangent and if of a being this one here and then a would be this pi over four or, as we see here, this limit. So just a week Write that down. We can say f of eggs equals tangent of X and a Therefore it would equal pile before, So that's just some logistics. But it's basically saying that Tangent of X does indeed have a derivative and we're going to use the room 3.12 in the textbook that help us solve this. And theory, three point told is just giving us derivatives of various trig functions. So if we have de de X, which is another way to write derivative of tangent of X theorem, 3.12 says that the derivative of tangent of X C can't squared of X and now we want to plug in Hi. Over four. Plug in. Hi. Over four. So we're gonna plug in pi over four into this X, so we'll get c Can't squared Opie over four. And it would be a good idea to pull up your unit circle for this. So pi over four is 45 degrees. So roughly there me at the values radical to over to radical to over two. And remember that you can't is the reciprocal co sign and co sign is always our X value. So what we do is take at X value and flip it. Remember, we can't have radicals and the denominator, so we have to rationalize and to rationalize. We multiply the numerator and denominator a radical to over to. So we'll get to radical too. And when you multiply to radicals together, they cancel out and just leave you what's left inside. And these 22 years will cancel. So we get radical too. And now we're gonna plug this back into our C can equation. So we get, um, radical, too. Over two squared. And when you're squaring a radical, you just get to which is our final answer.

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