How do you find limits of piecewise functions?

Question

I have a problem on limits. The graph shows two very different equations approaching the same point, but at that point (x=4) there is a hole. Below the hole there is a point at (4,2). What is the limit from the right hand going to 4 (lim x-> 4^+), what is the limit from the left hand going to 4 (lim x->4^-), and what is the limit as x approaches four (lim x -> 4)? Note that the final question is not what is f(x), but what is the limit as x approaches 4. Thank you!

f(x) = { 2x-2, 0 ≤ x <4,                           2, x=4,                   (x^2) -8x + 22, 4 <x

Answer 1

They are setting you up to learn about derivatives.  In order for a derivative to exist at some point (like x=4), the limit from the left must equal the limit from the right and they both must equal the limit at the point.  So the hole is a give away that there will be a problem.

 

Usually, as you approach where you want to know the limit, you look at the y value.

 

I hope this is a bit clearer!

Answer 2

Lim(f(x)) x->4- = 6

Lim(f(x)) x->4+ = 6  

 

so Lim(f(x)) x->4 = 6.

 

Yes, the function has a discontinuity at x=4 but the limit as x->4 is 6  (as x approaches 4, f(x) approaches 6 from both sides).  

 

The discontinuity at x=4 is called a "removable discontinuity" because many times the function can be patched up to exclude the discontinuity at the individual point(s).