Practice Limits and continuity
A limit describes the value a function approaches; continuity means its value matches that limit at a point.
These ideas underpin derivatives and integrals.
Exercises with solutions
A limit describes the value a function approaches as x tends to a point or to infinity. A function is continuous at a point when the left-hand and right-hand limits agree and both equal the function value. Points where the function is undefined are typical places of discontinuity.
Exercise 1
Find the limit of f(x) = 3x + 1 as x tends to 2.
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7
For a polynomial you may substitute directly: 3 · 2 + 1 = 7.
Exercise 2
Find the limit of f(x) = 1/x as x tends to infinity.
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0
The larger x becomes, the smaller 1/x becomes. The value approaches 0.
Exercise 3
Where is f(x) = 1/(x − 3) undefined?
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at x = 3
At x = 3 the denominator becomes zero. That is where the function is undefined.
Exercise 4
What does it mean when the left-hand and right-hand limits differ?
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the function is discontinuous there
Without agreeing one-sided limits there is no limit at that point, and the function is discontinuous there.
Common mistake
Confusing the function value with the limit. The limit describes behaviour near a point; at a point where the function is undefined there is no function value at all.
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