Find the limit should it exist.

Question

f(x) = 6 - x, if x < 4

4, if x = 4

x/4, if x > 4

a) limit of f(x) as x approaches 2 from the positive side

b) limit of f(x) as x approaches 2

c) limit of f(x) as x approaches 2 from the negative side

Answer 1

Hi there!

It's interesting that your function changes twice around the point, x=4, and your question asks about x=2. Thankfully, that means we can entirely work with the x<4 portion!

The simplest way to find your answers will be to plug in values increasingly close to 2 on each side. Just make a table, maybe 5-7 entries large.

x f(x), which is 6-x each time for being less than 4.

1.99 4.01

1.9999 4.0001

2 4

2.0001 3.9999

2.01 3.99

It appears that approaching from the negative side (barely less than 2), the limit is 4 on the right side of 4. Meanwhile, it appears that approaching from the positive side (barely greater than 2), the limit is 4 on the left side of 4. And in fact, because the limits on both sides equal 4, the limit from both sides is equal to 4 as well!

Caution: finding the limit as x approaches 2 is not as simple as just plugging in the value. Notice that the limit as x=4 would not exist as from the left it would be 2, right on would be 4, and from the right would be 1.

Side note: while the question is only asking for limits, we can also see that around x=2, the function is continuous because f(2) = 4, which is also its limit. This may feel obvious for simple functions, but understanding it now lays the groundwork for trickier problems.

ALTERNATIVELY,

you can find the limit from each side of 2 by setting x equal to 2+h (positive side) and 2-h (negative side) and setting the h limit to 0.

Thank you, and if you have any questions, please reach out!