anonymous 2 years ago Calc, help! How do I determine at what points y=x+2/cos x is continuous?

1. anonymous

where the function is defined

2. anonymous

what's the def of continuous?

3. anonymous

not at asymptote

4. anonymous

its not defined in text so yeah im assuming

5. anonymous

Definition in terms of limits of functions The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c through the domain of f exists and is equal to f(c).[2] In mathematical notation, this is written as $\lim_{x \to c}{f(x)} = f(c).$ In detail this means three conditions: first, f has to be defined at c. Second, the limit on the left hand side of that equation has to exist. Third, the value of this limit must equal f(c).

6. anonymous

as x approaches what value is the function undefined

7. anonymous

take the derivative

8. anonymous

can you walk me through that step by step if possible? I missed the lecture and am somewhat lost

9. anonymous

you need to use the quotient rule and the derivative of cos is -sin

10. anonymous

f(x)/g(x)=l/m?

11. anonymous

the function is not defined when $x=\frac{ \pi }{ 2 }+k\pi\text{, where }k \in \mathbb{Z}$

12. anonymous

because cos x will be 0 at those values of x and the function will not be deifned there. thus, the function will be discontinuous at those points

13. anonymous

so what you have above pi/2+ k(pi), how did you determine points off that?

14. anonymous

y = x is continuous for all real x. y = 2/x is continuous for all real x where x is not 0. -1 <= cos x <= 1 for all real x. thus, so long as cos x not = 0, your function will be continuous.

15. anonymous

ohhhhh. Ok I got it now. Thank you!