42. The limit of the function
(1 + 1/n)n as n → ∞ is
(A) ln x
(B) ln(1/x)
(C) e−x
(D) ex
Limit of (1 + 1/n)n as n → ∞ Equals e
The limit
limn→∞ (1 + 1/n)n
equals the mathematical constant
e ≈ 2.71828.
This result plays a foundational role in calculus and exponential growth.
Hence, among the given options, the correct match is
option (D) ex
when evaluated at x = 1, even though x is not explicitly present in the original expression.
Proof Using Logarithms
Taking natural logarithm:
ln y = limn→∞ n · ln(1 + 1/n)
This is an indeterminate form ∞ · 0.
Rewrite it as:
ln(1 + 1/n)/1/n
This becomes the 0/0 form. Using L’Hôpital’s Rule
or the Taylor expansion:
Hence:
Binomial Expansion Approach
Simplifying:
n(n−1)…(n−k+1)/nk
As n → ∞, each term approaches 1/k!.
Therefore:
Option Analysis
Logarithmic and unbounded. As x → 0+, ln x → −∞.
Does not match the constant value e.
Equivalent to −ln x, also unbounded.
Not suitable for a finite constant limit.
Represents exponential decay.
At x = 1, value = 1/e ≈ 0.367.
Note: limn→∞ (1 − 1/n)n = e−1.
At x = 1, e1 = e.
This directly matches the given limit.
Correct Answer.
Key Properties
- Monotonically increasing sequence
- Bounded above by e
- Converges strictly to e
Generalization
Common Misconceptions
Many assume the form 1∞ = 1
because 1/n → 0. However, the exponent n subtly amplifies the base,
leading to the emergence of e.
Options involving logarithms depend on x and cannot represent a fixed constant.
