How to determine the Left and Right limit of a function

[kmunzenmaier]

Hi everyone,

I'm having a lot of trouble with this question on my instructor's Midterm Review; I know how to determine left-hand and right-hand limits when the value of a is given (the value that x approaches), but we haven't been given this type of problem in the homework or the textbook for finding the left and right limit of a function without any value of a being given. Are there any great examples of how to solve a problem of this type when the value of a isn't provided for determining the left and right limit of a function?


Based on the given function for Part a), I know that the function is undefined when x=2, but the left and right limit could still exist at this point despite the discontinuity; is there a method/formula I'm missing for determining the left and right limit generally for the function (I have no idea how to find left and right limits without a given value of a)

Part b) is also very weird to me since we've never been given anything like this in the homework or the textbook (I made a differentiability checklist to the left of Part b) showing the "rules" but I'm still not fully understanding how to find the values of m and b based on differentiability ~ the homework we're given is completely different from the Midterm Review questions so I'm pretty stumped when it comes to these types of questions) It feels like Part b) might be a trick question though since the domain of f(x)=mx+b in the piecewise-defined function is limited to the domain of all x values greater than 2 :/

Any help is appreciated ~ thank you

 

[Steven G]

You are correct. You can not find the left and/or right hand limit unless you are given two things--one is the function and the other is the value that x is approaching.

For part a they must mean find the limits as x approaches 2 but they need to say this. I would get quick points for that question on an exam by just saying it can't be done.

 

[Steven G]

Why is part b strange. Did you graph the function? Please post your graph.

To be differentiable at x=a, the function must be continuous at x=a among other criterions.

 

[kmunzenmaier]

Why is part b strange. Did you graph the function? Please post your graph.

To be differentiable at x=a, the function must be continuous at x=a among other criterions.

Hi Jomo,

Thank you for giving me feedback for my question ~ Here's a photo of my graph for the entire function (I was also able to solve Part a) with x approaching 2):



Looking at my graph, I think my answer isn't a bad guess; I let m=1 and b=2 for the original function piece of of f(x)=mx+b if x>2, and this allowed the function to be continuous at x=2 which means it would also be differentiable at x=2 ~ I went through my continuity checklist and made sure that f(2) is defined, that the limit exists where the left-hand and right-hand limit must be the same, and that the limit as x approaches 2 equals f(2). I think my m and b values are correct but I'm still unsure; thank you for the help!

 

[Steven G]

Clearly your curve is not differentiable at x=2. It has a corner point at x=2, so it can't be differentiable at x=2. You need to change the slope of the line to remove that corner point.

 

[Steven G]

No, continuity does NOT imply differentiability!
Actually, it is the other way around. Differentiability implies continuity.

Recall that a limit exists if and only if the left hand limit equals the right hand limit. Also recall that a derivative is a limit. Hence the left hand derivative must equal the right hand derivative for a derivative to exist. In your graph it does not. The left hand derivative at x= 2 is 2x=2*2 = 4 while the right hand derivative is m=1.

 

[Steven G]

For part a, which function did you use to compute the right hand limit?