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

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→∞

Limit of sinx/x when x→∞

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Limit of sin(1/x) when x→0

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