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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.

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Chapter 3

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Chapter 3

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Chapter 3

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Chapter 3

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Chapter 3

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