- Topic
- Continuity
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Continuity practice question 8
The graph of a function is shown, with dashed asymptotes at and . Use it to find the one-sided limit below.
The small plus sign in says which side comes from. Which branch of the graph does that pick out?
means approaches from the right, through values such as and . Follow the branch to the right of the dashed line as it closes in on it.
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
A one-sided limit uses only one branch of the graph. Find the branch just right of x = −2 and see what its heights do as it closes in on the dashed line.
Working
Approaching from the right means using the branch to the right of the dashed line.
As that branch closes in on x = −2 it drops out of the bottom of the window and keeps falling, without bound.
The left branch climbs without bound instead, so the side matters. Either way h(−2) is undefined, and h has an infinite discontinuity at −2.
Answer
The answer is negative infinity.
Check
The right branch rises through (−1, −1) away from the asymptote, so followed back towards x = −2 it falls, out of the bottom of the window.
Where this goes wrong
Reading the branch on the wrong side of the asymptote and answering +∞. The plus in −2⁺ says x is greater than −2, which is the branch to the right of the dashed line.
Leans on
Infinite limits, from Limits
Try a similar question
Evaluate the one-sided limit.Saved in this browser only.