The limit of (1+1/x)^x as x approaches infinity is equal to e. Here we will discuss Lim x→∞ (1+(1/x))^x formula with proof.

Note that

Lim_{x → ∞} (1+$\frac{1}{x}$)^{x} = e.

## Lim_{x→∞} (1+(1/x))^{x} Formula with Proof

The formula of lim_{x→∞} (1+(1/x))^{x} is given by lim_{x → ∞} (1+(1/x))^{x} =e.

**Explanation:**

Let y = lim_{x→∞} (1+$\frac{1}{x}$)^{x}.

So we need to find y.

Taking natural logarithm ln on both sides, we get that

ln y = ln lim_{x→∞} (1+$\frac{1}{x}$)^{x}

⇒ ln y = lim_{x→∞} ln (1+$\frac{1}{x}$)^{x}

⇒ ln y = lim_{x→∞} x ln(1+$\frac{1}{x}$) as we know ln lim_{x→∞} f(x) = lim_{x→∞} ln f(x).

Take 1/x = z.

Then z→0 as x→∞. So from above we get that

ln y = lim_{z→0} $\dfrac{\ln (1+z)}{z}$

⇒ ln y = 1 using the formula lim_{x→0} $\dfrac{\ln (1+x)}{x}$ =1.

⇒ ln y = ln e

⇒ y = e.

In other words, lim_{x→∞} (1+$\frac{1}{x}$)^{x} =e.

So the limit of (1+1/x)^{x} as x approaches Infinity is equal to e.

**Read Also:** Limit of cosx/x when x→∞

## FAQs

### Q1: What is the limit of (1+1/x)^{x} when x tends to infinity?

Answer: The limit of (1+1/x)^{x} when x tends to infinity is equal to e, that is, lim_{x → ∞} (1+1/x)^{x} = e.

### Q2: What is Lim_{x → ∞} (1+(1/x))^{x}?

Answer: Lim_{x → ∞} (1+(1/x))^{x} = e.