Finding the Limit of (1 + (1/x))^x as x Approaches Infinity | Step-By-Step Analysis

lim x-> ∞ (1 + (1/x)) ^x

To find the limit of the given expression as x approaches infinity, let’s analyze it step by step

To find the limit of the given expression as x approaches infinity, let’s analyze it step by step.

The expression is:

lim x-> ∞ (1 + (1/x))^x

Let’s start by recognizing that the expression is of the indeterminate form “∞^0”. Therefore, one approach is to convert it into the form “e^(∞ * ln(1 + (1/x)))”.

Next, we can use the fact that ln(a^b) = b * ln(a) to simplify the expression:

= lim x-> ∞ e^(x * ln(1 + (1/x)))

Now, let’s consider the term ln(1 + (1/x)). As x approaches infinity, 1/x approaches zero. So, we can replace 1/x with 0 in the expression:

= lim x-> ∞ e^(x * ln(1 + 0))

Since ln(1) is equal to 0, we have:

= lim x-> ∞ e^0

Remember that e^0 is equal to 1. Therefore, the final result is:

= 1

Hence, the limit of (1 + (1/x))^x as x approaches infinity is 1.

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