As x approaches a, approaching from the larger or smaller side is called the right-hand limit or left-hand limit, respectively.
One-Sided Limits
- Right-hand limit: lim_{x→a+0} f(x)
- Left-hand limit: lim_{x→a−0} f(x)
A limit exists if and only if the right-hand and left-hand limits are equal.
Example ①
For f(x) = |x|/x, investigate lim_{x→0}.
Solution:For x>0, it is 1; for x<0, it is −1. The right-hand limit is 1 and the left-hand limit is −1. Since they are not equal, the limit does not exist.
For f(x) = |x|/x, investigate lim_{x→0}.
Solution:For x>0, it is 1; for x<0, it is −1. The right-hand limit is 1 and the left-hand limit is −1. Since they are not equal, the limit does not exist.
Limits at Infinity
The same idea applies to x → ∞ and x → −∞. lim_{x→∞} (3x²+1)/(x²+x) = 3
Divergence as x → a
lim_{x→0+} 1/x = ∞, lim_{x→0−} 1/x = −∞
Example ②
lim_{x→∞} (√(x²+1) − x)
Solution:Rationalizing: 1/(√(x²+1)+x) → 0
lim_{x→∞} (√(x²+1) − x)
Solution:Rationalizing: 1/(√(x²+1)+x) → 0
Key Points
- A limit exists only when the left-hand and right-hand limits are equal.
- For |x|/x and step functions, the two one-sided limits differ.
- x → ∞ is handled in the same way as for sequences.
- 1/(x−a) diverges to ±∞ as x→a.
Practice Problems
- lim_{x→2+0} 1/(x−2)、lim_{x→2−0} 1/(x−2)
- lim_{x→∞} (2x²−x+1)/(x²+3)
- lim_{x→−∞} (√(x²+1) + x)
Answers and Explanations
- Answer:Right-hand ∞, left-hand −∞
The behavior differs according to the sign. - Answer:2
Divide by x². - Answer:0
Let t=−x>0 and rationalize.