Answer to Question #131054 in Calculus for Break Down

Question #131054

Ali is trying to find the limit of a function which is expressed as L = lim xβ†’0+ (1 + x)1/x . From his understanding, the quantity (1 + x), must be greater than 1 for x > 0. Furthermore, the power1 / x is going to infinity as ? approaches 0 from the right. So L is the result of taking a number greater than 1 to higher power, therefore L = ∞. On the other hand, he sees that (1 + x) is approaching 1 as x approaches 0, and 1 taken to any power whatever is 1. Ali concluded, L = 1. Help Ali by pointing out to him the error of his ways.


1
Expert's answer
2020-08-31T17:15:34-0400

Let xn=1n,n∈Z+x_n=\dfrac{1}{n}, n\in\Z^+


lim⁑xnβ†’0+(1+xn)1xn=lim⁑nβ†’βˆž(1+1n)n\lim\limits_{x_n\to 0^+}(1+x_n)^{{1 \over x_n}}=\lim\limits_{n\to \infin}(1+{1\over n})^{n}

Use Newton's Binomial formula

(1+1n)n=1+nβ‹…1n+n(nβˆ’1)1β‹…2β‹…1n2+(1+{1\over n})^{n}=1+n\cdot{1\over n}+{n(n-1)\over 1\cdot2}\cdot{1\over n^2}+

+n(nβˆ’1)(nβˆ’2)1β‹…2β‹…3β‹…1n3+...+n(nβˆ’1)...(1)1β‹…2β‹…...β‹…nβ‹…1nn>1+1=2+{n(n-1)(n-2)\over 1\cdot2\cdot3}\cdot{1\over n^3}+...+{n(n-1)...(1)\over 1\cdot2\cdot...\cdot n}\cdot{1\over n^n}>1+1=2



Let f(x)=ln⁑(x),f(x)=\ln (x), then fβ€²(x)=1/x.f'(x)=1/x. Thus fβ€²(1)=1/1=1.f'(1)=1/1=1.

From the definition of a derivative as a limit, we have


fβ€²(1)=lim⁑hβ†’0f(1+h)βˆ’f(1)h=f'(1)=\lim\limits_{h\to 0}\dfrac{f(1+h)-f(1)}{h}=

=lim⁑xβ†’0f(1+x)βˆ’f(1)x=lim⁑xβ†’0ln⁑(1+x)βˆ’ln⁑(1)x==\lim\limits_{x\to 0}\dfrac{f(1+x)-f(1)}{x}=\lim\limits_{x\to 0}\dfrac{\ln(1+x)- \ln(1)}{x}=

=lim⁑xβ†’0ln⁑(1+x)x=lim⁑xβ†’0ln⁑(1+x)1x=\lim\limits_{x\to 0}\dfrac{\ln(1+x) }{x}=\lim\limits_{x\to 0}\ln(1+x)^{{1\over x}}

Because fβ€²(1)=1,f'(1)=1, we have


lim⁑xβ†’0ln⁑(1+x)1x=1\lim\limits_{x\to 0}\ln(1+x)^{{1\over x}}=1

Theorem

If ff is continuous at bb and lim⁑xβ†’ag(x)=b,\lim\limits_{x\to a}g(x)=b, then lim⁑xβ†’af(g(x))=f(b)\lim\limits_{x\to a}f(g(x))=f(b)


lim⁑xβ†’af(g(x))=f(lim⁑xβ†’ag(x))\lim\limits_{x\to a}f(g(x))=f(\lim\limits_{x\to a}g(x))

Since the exponential function is continuous, we have


e=e1=elim⁑xβ†’0ln⁑(1+x)1x=lim⁑xβ†’0eln⁑(1+x)1x=e=e^1=e^{\lim\limits_{x\to 0}\ln(1+x)^{{1\over x}}}=\lim\limits_{x\to 0}e^{\ln(1+x)^{{1\over x}}}=

=lim⁑xβ†’0(1+x)1x=\lim\limits_{x\to 0}(1+x)^{{1\over x}}

Therefore


lim⁑xβ†’0(1+x)1x=e\lim\limits_{x\to 0}(1+x)^{{1\over x}}=e

lim⁑xβ†’0+(1+x)1x=e\lim\limits_{x\to 0^+}(1+x)^{{1\over x}}=e


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