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Problem 65

# Show that the exponential function defined by $$\exp \mathrm{x}=\lim _{\mathrm{n} \rightarrow \infty}[1+(\mathrm{x} / \mathrm{n})]^{\mathrm{n}}$$ is everywhere continuous.

Expert verified
To show that the exponential function is everywhere continuous, we first expand the expression $$(1 + \frac{x}{n})^n$$ using the binomial theorem, and rewrite the limit as: $$\lim_{n \to \infty} (1 + \frac{x}{n})^n = \lim_{n \to \infty} \sum_{k=0}^n \binom{n}{k} \frac{x^k}{n^k}$$ By examining the limit and using the relationship between exp(x) and the natural logarithm ln(x), we have: $$\lim_{x \to 0} \ln(\lim_{n \to \infty} (1 + \frac{x}{n})^n) = \lim_{x \to 0} \ln(\exp(x))= \lim_{x \to 0} x$$ Since the limit exists and equals the function value as x approaches 0, we conclude that the exponential function is continuous at every point x.
See the step by step solution

## Step 1: 1. Existing limit

: To check if the limit exists at every point x, we first expand the expression $$(1 + \frac{x}{n})^n$$ using the binomial theorem: $$(1 + \frac{x}{n})^n = \sum_{k=0}^n \binom{n}{k} \frac{x^k}{n^k}$$, where $$\binom{n}{k}$$ is the binomial coefficient. Now, let's examine the limit: $$\lim_{n \to \infty} (1 + \frac{x}{n})^n = \lim_{n \to \infty} \sum_{k=0}^n \binom{n}{k} \frac{x^k}{n^k}$$, We will need to find the (finite) limit of this expression as n approaches infinity.

## Step 2: 2. Continuous function at every point x

: To establish that the exponential function is continuous at every point x, let's take a closer look at the limit. We can rewrite the exponential function as: $$\exp(x) = \lim_{n \to \infty} (1 + \frac{x}{n})^n$$, By examining the limit: $$\lim_{x \to 0} \frac{\exp(x) - 1}{x} = \lim_{x \to 0} \frac{\lim_{n \to \infty} (1 + \frac{x}{n})^n - 1}{x}$$, Now, by using the relationship between exp(x) and the natural logarithm ln(x): $$\lim_{x \to 0} \ln(\lim_{n \to \infty} (1 + \frac{x}{n})^n) = \lim_{x \to 0} \ln(\exp(x))= \lim_{x \to 0} x$$, We observe that as x approaches 0, the limit exists and equals the function value, thus establishing that the exponential function is continuous at every point x.

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