One-Sided Limits and When a Limit Fails to Exist
Compare one-sided behavior. Matching finite limits give a finite two-sided limit. Shared growth toward positive or negative infinity is called an infinite limit, not a finite limit. Disagreement or persistent failure to settle prevents a shared two-sided limit.
What a learner can do afterwards
- Compare supplied one-sided behavior and interpret estimates suggested by tables
- Analyze piecewise formulas and determine whether their one-sided limits agree
- Distinguish finite limits, shared infinite directions, disagreement and persistent oscillation
1 · Read
A right-hand limit uses inputs greater than the approach point. For a positive number u, |u| = u, so |u|/u = 1. Use the formula valid on the right and find what its outputs approach. The assigned value at the approach point is not used.
A left-hand limit uses inputs smaller than the approach point. For a negative number u, |u| = -u, so |u|/u = -1. Choose the left branch before simplifying. A right-side formula need not describe the left-side behavior.
A finite two-sided limit exists exactly when both one-sided limits exist as finite numbers and agree. A finite table can suggest an estimate, but cannot prove an exact limit: unsampled inputs may behave differently. Read each table toward the approach point and compare the suggested targets.
For f(x) = x+2 below 5 and 12-x above 5, the branch limits are 5+2 = 7 and 12-5 = 7. Thus the limit is 7, whatever value is assigned at 5. For polynomial branches, substitute the approach point into each branch separately.
Unequal finite branch limits give no two-sided limit. Averaging them does not create agreement. Changing only f(c) cannot change either branch limit. To choose a parameter that makes branches agree, equate their limiting values and solve. For 3x below 1 and x+k above 1, agreement requires 3 = 1+k, so k = 2.
An infinite limit is not a finite limit. It describes both sides tending to the same positive or negative infinite direction. In A/(x-c)², the denominator stays positive and tends to zero. Positive A gives positive infinity on both sides; negative A gives negative infinity on both sides.
For 1/(x-c), the denominator changes sign: the left tends to negative infinity and the right to positive infinity. There is no shared two-sided infinite direction. Also 1/|x-c| is positive on both sides. Simplify combinations of these expressions on each side before comparing their behavior.
A finite limit on one side and an infinite direction on the other do not agree. Persistent oscillation can also prevent a one-sided limit from settling, even if the other side has a limit. For reciprocals, outputs tending to positive infinity have reciprocals tending to zero; outputs tending to a nonzero finite B have reciprocals tending to 1/B.
Compare both sides. Distinguish a shared finite value, a shared infinite direction, disagreement and failure to settle.
Original one-sided-limit lesson, questions and flashcards written for this learning dataset.
Practice questions written for this learning dataset. The reading paraphrases When Does a Limit Not Exist?; its source license is unspecified. Dataset flashcards were revised.
2 · Watch
به انگلیسیTake it off screen
Where it sits
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.