40. Let a = (√5 + 1)/2 and b = (√5 − 1)/2. Then, evaluate:
limn→∞ (an + bn)/(an − bn)
(A) is 1
(B) is 1/2
(C) is 0
(D) does not exist
Evaluate the Limit of (aⁿ + bⁿ)/(aⁿ − bⁿ) as n Tends to Infinity
Understanding the Given Limit Problem
This question is based on the behaviour of exponential terms as n tends to infinity. At first sight, the expression may appear difficult because both the numerator and denominator contain powers of two irrational numbers. However, the problem becomes simple once we compare the magnitudes of a and b.
The given values are:
a = (√5 + 1)/2
and
b = (√5 − 1)/2
We need to evaluate:
limn→∞ (an + bn)/(an − bn)
The main idea is to determine which exponential term dominates as n becomes very large.
Comparing the Values of a and b
Using the approximate value:
√5 ≈ 2.236
we obtain:
a = (2.236 + 1)/2 ≈ 1.618
and:
b = (2.236 − 1)/2 ≈ 0.618
Therefore:
a > 1
whereas:
0 < b < 1
This comparison is important because when a positive number greater than 1 is raised to increasingly large powers, its value increases. In contrast, when a positive number lying between 0 and 1 is raised to increasingly large powers, its value approaches zero.
Hence, as n → ∞:
an → ∞
and:
bn → 0
This already suggests that the terms containing an will dominate both the numerator and denominator.
Simplifying the Limit by Dividing by the Dominant Term
The given limit is:
limn→∞ (an + bn)/(an − bn)
Since an is the dominant term, we divide both the numerator and denominator by an. This gives:
limn→∞ [(an/an) + (bn/an)] / [(an/an) − (bn/an)]
After simplification:
limn→∞ [1 + (b/a)n] / [1 − (b/a)n]
Therefore, the entire problem now depends on the behaviour of (b/a)n as n tends to infinity.
Finding the Value of b/a
Using the given values:
b/a = [(√5 − 1)/2] / [(√5 + 1)/2]
The common factor 1/2 cancels, giving:
b/a = (√5 − 1)/(√5 + 1)
Rationalizing the denominator:
b/a = [(√5 − 1)(√5 − 1)] / [(√5 + 1)(√5 − 1)]
Therefore:
b/a = (√5 − 1)²/(5 − 1)
Expanding the numerator:
(√5 − 1)² = 5 − 2√5 + 1 = 6 − 2√5
Hence:
b/a = (6 − 2√5)/4 = (3 − √5)/2
Numerically:
b/a ≈ 0.382
Thus:
0 < b/a < 1
Applying the Standard Exponential Limit
For any real number r satisfying |r| < 1, the following standard result holds:
limn→∞ rn = 0
Since:
0 < b/a < 1
we get:
limn→∞ (b/a)n = 0
Substituting this result into the simplified expression:
limn→∞ [1 + (b/a)n] / [1 − (b/a)n]
we obtain:
(1 + 0)/(1 − 0)
Therefore:
1/1 = 1
Alternative Explanation Using Dominant Terms
The result can also be understood directly by comparing the terms in the original expression. Since a ≈ 1.618 and b ≈ 0.618, the term an becomes much larger than bn as n increases.
Therefore, for very large values of n, the numerator behaves approximately as:
an + bn ≈ an
Similarly, the denominator behaves approximately as:
an − bn ≈ an
Thus, the ratio approaches:
an/an = 1
This confirms the result obtained using the formal limit method.
Analysis of All the Given Options
Option (A): The Limit Is 1
This option is correct. After dividing the numerator and denominator by an, the expression becomes [1 + (b/a)n]/[1 − (b/a)n]. Since 0 < b/a < 1, the term (b/a)n approaches zero. Therefore, the limit becomes (1 + 0)/(1 − 0) = 1.
Option (B): The Limit Is 1/2
This option is incorrect. There is no factor in the simplified limit that produces the value 1/2. The dominant term an appears with the same coefficient in both the numerator and denominator, so their ratio approaches 1, not 1/2.
Option (C): The Limit Is 0
This option is incorrect. Although bn approaches zero, the entire numerator does not approach zero because it also contains an, which grows without bound. The numerator and denominator are both dominated by the same term an, causing their ratio to approach 1.
Option (D): The Limit Does Not Exist
This option is also incorrect. The ratio b/a has magnitude less than 1, so (b/a)n approaches zero in a well-defined manner. Consequently, the given expression approaches the definite finite value 1, and the limit exists.
Final Answer
Therefore:
limn→∞ (an + bn)/(an − bn) = 1
Correct Option: (A) is 1
