(A) 1
(B) 1/2
(C) 0
(D) Does not exist
Evaluate the Limit of (aⁿ + bⁿ)/(aⁿ − bⁿ) as n Approaches Infinity
Understanding the Given Limit Problem
This question asks us to evaluate the limiting value of a sequence containing two exponential terms, an and bn. The key to solving the problem is to compare the numerical values of a and b and determine which exponential term dominates as n becomes extremely large.
The given constants are:
a = (√5 + 1)/2
and:
b = (√5 − 1)/2
The expression whose limit is required is:
limn→∞ (an + bn)/(an − bn)
At first glance, both the numerator and denominator contain terms raised to the power n, which may make the expression appear complicated. However, the values of a and b are very different in their long-term exponential behavior. The number a is greater than 1, while b lies between 0 and 1. Therefore, an grows without bound, whereas bn approaches zero.
This difference in growth behavior is the central idea required to evaluate the limit.
Calculate the Approximate Values of a and b
To understand the behavior of the sequence, we first examine the approximate numerical values of a and b. Since:
√5 ≈ 2.236
we obtain:
a = (2.236 + 1)/2
Therefore:
a ≈ 1.618
Similarly:
b = (2.236 − 1)/2
Therefore:
b ≈ 0.618
Thus, we have:
a > 1
and:
0 < b < 1
This immediately tells us that the two exponential terms behave differently as n approaches infinity.
Behavior of aⁿ and bⁿ as n Approaches Infinity
Because a ≈ 1.618 is greater than 1, repeated multiplication causes an to become larger and larger. Therefore:
an → ∞ as n → ∞
On the other hand, b ≈ 0.618 lies between 0 and 1. When a positive number smaller than 1 is repeatedly multiplied by itself, its powers become progressively smaller.
Therefore:
bn → 0 as n → ∞
This means that for very large values of n, the term an dominates both the numerator and denominator, while the contribution of bn becomes comparatively negligible.
Step-by-Step Solution
Step 1: Write the Required Limit
The given limit is:
L = limn→∞ (an + bn)/(an − bn)
where:
a = (√5 + 1)/2
and:
b = (√5 − 1)/2
Since a is greater than b, the most effective method is to divide both the numerator and denominator by the dominant exponential term an.
Step 2: Divide the Numerator and Denominator by aⁿ
Dividing every term in the numerator and denominator by an gives:
L = limn→∞
[(an/an) + (bn/an)]
/
[(an/an) − (bn/an)]
Since:
an/an = 1
and:
bn/an = (b/a)n
the limit becomes:
L = limn→∞
[1 + (b/a)n]
/
[1 − (b/a)n]
The original problem has now been reduced to finding the behavior of the ratio (b/a)n.
Step 3: Calculate the Ratio b/a
Using the given values:
b/a = [(√5 − 1)/2] / [(√5 + 1)/2]
The factors of 2 cancel, giving:
b/a = (√5 − 1)/(√5 + 1)
Rationalizing the denominator:
b/a = [(√5 − 1)(√5 − 1)]/[(√5 + 1)(√5 − 1)]
The denominator becomes:
5 − 1 = 4
The numerator becomes:
(√5 − 1)2 = 6 − 2√5
Therefore:
b/a = (6 − 2√5)/4
Thus:
b/a = (3 − √5)/2
Numerically:
b/a ≈ 0.382
Since this value lies strictly between 0 and 1, its nth power approaches zero as n approaches infinity.
Step 4: Evaluate the Limit of (b/a)ⁿ
We have established that:
0 < b/a < 1
A fundamental result for exponential sequences states that if |r| < 1, then:
rn → 0 as n → ∞
Therefore:
limn→∞ (b/a)n = 0
This is the decisive step in the solution.
Step 5: Substitute the Limiting Value
The simplified expression is:
L = limn→∞
[1 + (b/a)n]
/
[1 − (b/a)n]
Since:
(b/a)n → 0
we obtain:
L = (1 + 0)/(1 − 0)
Therefore:
L = 1/1
Hence:
L = 1
Complete Solution in Compact Form
The required limit is:
limn→∞ (an + bn)/(an − bn)
Dividing the numerator and denominator by an:
= limn→∞
[1 + (b/a)n]
/
[1 − (b/a)n]
Since:
0 < b/a < 1
we have:
limn→∞ (b/a)n = 0
Therefore:
= (1 + 0)/(1 − 0)
= 1
Alternative Explanation Using the Dominant Term Method
The answer can also be understood without performing the full algebraic simplification. Since a ≈ 1.618 and b ≈ 0.618, the term an becomes extremely large as n increases, while bn becomes extremely small.
Therefore, for sufficiently large n:
an + bn ≈ an
and:
an − bn ≈ an
Hence, the ratio behaves like:
an/an
which is equal to:
1
This dominant term approach gives the same result and provides an intuitive explanation of why the sequence approaches 1.
Special Relationship Between a and b
The given numbers have an interesting algebraic relationship. Multiplying a and b gives:
ab = [(√5 + 1)/2][(√5 − 1)/2]
Using the difference of squares:
ab = (5 − 1)/4
Therefore:
ab = 1
Hence:
b = 1/a
As a result:
b/a = 1/a2
Since a > 1:
0 < 1/a2 < 1
Therefore:
(b/a)n = (1/a2)n → 0
This provides another direct justification for the final limit.
Why the Limit Approaches 1 from Above
For every positive integer n, both a and b are positive. Therefore:
an + bn > an − bn
Hence, the ratio:
(an + bn)/(an − bn)
is greater than 1 for finite positive values of n. However, as n increases, bn becomes negligible compared with an. The difference between the numerator and denominator therefore becomes proportionally smaller.
As a result, the sequence gets progressively closer to 1. Thus, the limiting value is exactly 1.
Detailed Analysis of Each Option
Option (A): 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): 1/2
This option is incorrect. The numerator and denominator are both dominated by the same leading term an. Therefore, their ratio approaches the ratio of their dominant coefficients, which is 1/1 = 1, not 1/2.
There is no factor in the expression that causes the numerator to become half of the denominator as n approaches infinity.
Option (C): 0
This option is incorrect. Although bn approaches zero, the complete numerator does not approach zero because it also contains an. In fact, an grows without bound.
Both the numerator and denominator are dominated by an, so their ratio approaches 1 rather than 0.
Option (D): Does Not Exist
This option is incorrect. The ratio b/a has magnitude less than 1, so (b/a)n converges to zero. Therefore, the entire sequence converges to the definite finite value 1.
The denominator also remains positive for positive integers n because a > b > 0, which gives an > bn. Hence, there is no oscillation or undefined long-term behavior that would prevent the limit from existing.
Verification Using Numerical Values
Using the approximate values:
a ≈ 1.618
and:
b ≈ 0.618
we can observe the behavior of the ratio. As n increases, an rapidly becomes much larger than bn. Consequently, adding or subtracting the very small term bn makes less and less difference compared with the dominant term an.
Thus:
(an + bn)/(an − bn) → 1
This numerical behavior fully agrees with the algebraic solution.
Final Answer
The value of limn→∞ (an + bn)/(an − bn), where a = (√5 + 1)/2 and b = (√5 − 1)/2, is 1.
Correct Option: (A) 1
