The first topic in Mathematics III is limits. We begin by examining the value that the sequence {a_n} approaches as n becomes arbitrarily large. This is the limit of a sequence.
Convergence and Divergence
When a sequence a_n approaches a value α as n→∞, we say that a_n converges to α and write
lim_{n→∞} a_n = α. α is called the limit value.
When a sequence does not converge, we say that it diverges. There are three patterns of divergence.
- Divergence to positive infinity: lim_{n→∞} a_n = ∞
- Divergence to negative infinity: lim_{n→∞} a_n = −∞
- Oscillation: It approaches neither ±∞ nor any particular value (example: a_n = (−1)^n)
Determine the limits of the following sequences.
(1) a_n = 1/n (2) a_n = n² (3) a_n = (−1)^n
Solution:
(1) As n increases, 1/n approaches 0 → lim = 0 (converges)
(2) As n increases, n² becomes arbitrarily large → lim = ∞ (diverges to positive infinity)
(3) It alternates between +1 and −1 and does not approach a particular value → oscillation (divergence)
Basic Limit Formulas
When lim_{n→∞} a_n = α and lim_{n→∞} b_n = β:
- lim (a_n ± b_n) = α ± β
- lim (k · a_n) = k · α (k is a constant)
- lim (a_n · b_n) = α · β
- lim (a_n / b_n) = α / β (provided β ≠ 0)
These can be used when α and β are finite values. For indeterminate forms such as ∞ − ∞ and ∞/∞, they cannot be applied directly, so algebraic transformations are necessary.
Limits of the ∞/∞ Form
For a fraction whose numerator and denominator both become ∞, the standard method is to divide the numerator and denominator by the highest-degree term.
Example: Consider lim_{n→∞} (3n² + 2n − 1)/(n² + 5). Dividing the numerator and denominator by n² gives
= lim_{n→∞} (3 + 2/n − 1/n²)/(1 + 5/n²) = (3 + 0 − 0)/(1 + 0) = 3
Find the following limit: lim_{n→∞} (2n² − 3n + 1)/(4n² + n + 5)
Solution: Divide the numerator and denominator by n².
= lim (2 − 3/n + 1/n²)/(4 + 1/n + 5/n²) = (2 − 0 + 0)/(4 + 0 + 0) = 1/2
- Convergence: lim_{n→∞} a_n = α (a finite value) / Divergence: ±∞ or oscillation
- 1/n → 0, 1/n² → 0, and the constant sequence c → c
- For the ∞/∞ form, divide the numerator and denominator by the highest power n^k
- For the ∞ − ∞ form, transform the expression by rationalization or factoring out common factors
Practice Problems
- Find lim_{n→∞} (5n + 3)/(2n − 1).
- Find lim_{n→∞} (n² + 1)/(n + 2).
- Find lim_{n→∞} (√(n² + n) − n).
Answers and Explanations
- Answer:5/2
Explanation: Divide the numerator and denominator by n. (5 + 3/n)/(2 − 1/n) → 5/2. - Answer:∞ (diverges to positive infinity)
Explanation: The degree of the numerator (2) is greater than the degree of the denominator (1), so the sequence diverges. - Answer:1/2
Explanation: Rationalize. n/(√(n²+n)+n). Dividing the numerator and denominator by n gives 1/(√(1+1/n)+1) → 1/2.