Limit of (1+1/x)^x as x approaches Infinity | Lim x→∞ (1+1/x)^x

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

Limx → ∞ (1+$\frac{1}{x}$)x = e.

Limx→∞ (1+(1/x))x Formula with Proof

The formula of limx→∞ (1+(1/x))x is given by limx → ∞ (1+(1/x))x =e.

Explanation:

Let y = limx→∞ (1+$\frac{1}{x}$)x.

So we need to find y.

Taking natural logarithm ln on both sides, we get that

ln y = ln limx→∞ (1+$\frac{1}{x}$)x

⇒ ln y = limx→∞ ln (1+$\frac{1}{x}$)x

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

Take 1/x = z.

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

ln y = limz→0 $\dfrac{\ln (1+z)}{z}$

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

⇒ ln y = ln e

⇒ y = e.

In other words, limx→∞ (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→∞

Limit of sinx/x when x→∞

Limit of cos(1/x) when x→∞

Limit of sin(1/x) when x→0

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, limx → ∞ (1+1/x)x = e.

Q2: What is Limx → ∞ (1+(1/x))x?

Answer: Limx → ∞ (1+(1/x))x = e.