Question

Whoa, whoa! What am I missing? Calc question.

Determine whether the function has a vertical asymptote or a removable discontinuity at x = −4. Graph the function using a graphing utility to confirm your answer.

Answer 1

\(\Large f(x) = \frac{\sin (x + 4)}{x+4}\)

Answer 2

So, if you plug in x = -4, you'll get 0 in the numerator and denominator which should make it undefined. Which should make the answer a vertical asymptote. But the correct answer is a removable discontinuity, which you can see if you graph it.
https://www.desmos.com/calculator/pyx6klrxac

But why is it a removable discontinuity and how is the function defined at x = -4?

Also, can you show me some way to do this that wouldn't rely using a graphing calculator? I'm not sure why the homework allows you to use it. We can't use any kind of calculator on the test or quiz.

Answer 3

\(\Large f(-4) = \sf undefined\)\[\Large \lim_{x \rightarrow -4} f(x) = 1\]
This is my understanding.

I don't see how f(-4) = 1
The limit as it approaches -4 can be 1, but the function cannot be defined at -4

Answer 4

Also, on a side note, we're currently learning about continuity of a function and derivatives, so I guess the question has a relation to it (:

Answer 5

@whpalmer4 Can you assist me? :)

Answer 6

This is one of your early limit identities that you learn about,\[\large\rm \lim_{x\to0}\frac{\sin x}{x}=1\]

Answer 7

Yeah, I learned that c:

Answer 8

Think about the numerator and denominator separately,