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A function is continuous at x = a when f(a) exists, lim(x→a) f(x) exists, and they are equal. Polynomials are continuous everywhere.
Limits at infinity tell end behavior. For rational functions, compare degrees: lim(x→∞) (3x² + 1)/(x² − 5) = 3.
Special limit: lim(x→0) sin x / x = 1 (x in radians).
Worked examples
Example 1
Find lim(x→∞) (4x − 7)/(2x + 9).
- Divide top and bottom by x: (4 − 7/x)/(2 + 9/x)
- As x → ∞ the small terms → 0
Answer: 2
Example 2
Is f(x) = x + 1 for x < 2 and 5 for x ≥ 2 continuous at x = 2?
- Left limit: 2 + 1 = 3
- Right limit and f(2): 5
- 3 ≠ 5
Answer: No, there is a jump
Practice
Try these on paper first, then check your work.
- lim(x→∞) 7/x²
- lim(x→∞) (x² + 1)/(3x²)
- lim(x→0) sin(x)/x
- Continuous at 0: f(x) = |x|?
Show the answer key
- 0
- 1/3
- 1
- Yes