Practice question · Put in order
Order the steps of testing whether f is continuous at the point a.
- Find the limit of f(x) as x approaches a from the right
- Check that f(a) is defined
- Check that walking in from either side leads to the same place
- Compare that limit with f(a) and declare continuity only if they match
- Find the limit of f(x) as x approaches a from the left
Hints
- You cannot compare the limit with the value until you have both of them.
- The two-sided limit is assembled out of the two one-sided limits.
Show the answer
- Check that f(a) is defined
- Find the limit of f(x) as x approaches a from the left
- Find the limit of f(x) as x approaches a from the right
- Check that walking in from either side leads to the same place
- Compare that limit with f(a) and declare continuity only if they match
Why
The value comes first, then each side separately, then their agreement, and only then the final comparison. Skipping straight to the last step is what makes students assume the limit and the value always coincide.
Practise Limits and Continuity
The app has 6 more questions on this lesson, and keeps your place in the course. Computer Science I is free to start.
More questions on Limits and Continuity
- Adding a million small floats into one running total loses accuracy the same numbers would keep if summed…
- Select every condition that must hold for f to be continuous at the point a.
- Calculus assumes exact real numbers and computers have neither exact reals nor infinitely many of them. Why…
- For each described point, sort by whether the two-sided limit exists there.