7

I'm reading this text:

enter image description here

A few questions:

  1. What's the importance of them going from $h$ to $x$ in the first line? What is the difference?

  2. How did they go from $$\lim_{x \to 0} \frac{\ln(1+x) - \ln(1)}{x}$$ to $$\lim_{x \to 0} \left[\frac{1}{x} \cdot \ln(1+x)\right]$$

  3. And then right before the blue 5 box... how did they go from: $$e^{\lim_{x \to 0} \ln(1+x)^{1/x}}$$ to $$\lim_{x \to 0} e^{\ln(1+x)^{1/x}}$$ How did they just pull out the limit sign?

eyeballfrog
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Jwan622
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2 Answers2

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The x-h notations are equivalent.

Note that $\ln 1=0$ thus

$$\lim_{x \to 0} \frac{\ln(1+x) - \ln(1)}{x}=\lim_{x \to 0} \left(\frac{1}{x} \cdot \ln(1+x)\right)$$

Since the exponential function is continuos $$e^{\lim f(x)}\equiv \lim e^{f(x)}$$

Botond
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user
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    when the e function is continuous, then e to the exponent is what? what does the three equals mean? – Jwan622 Feb 08 '18 at 22:29
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    @Jwan622 It basically means that $\lim\limits_{x\to a}f(g(x))=f(\lim\limits_{x\to a}g(x))$ if $f$ is continuous at $\lim\limits_{x\to a}g(x)$. So the $\lim$ and $f$ can "switch places". And the $\equiv$ means that they are equivalent. – Botond Feb 08 '18 at 22:32
  • @Jwan622 yes it is just as Botond stated! – user Feb 08 '18 at 22:37
  • @Botond And what does “equivalent” mean? The equality sign is better here. – egreg Feb 08 '18 at 22:39
  • @egreg it was just a symbol to describe that it is an equivalent way to take the limit – user Feb 08 '18 at 22:41
  • @egreg To be honest, I could not even draw it up on my mother tongue. – Botond Feb 08 '18 at 22:42
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    Equivalent is often used in situations where "equal" makes no sense. Equivalency calls for an extra structure that defines the equivlaency. For example, a square cannot equal a circle, but we can say "this square and this circle are equivalent in terms of area." In this case, "in terms of area" is the equivalency structure. Where it gets a bit strange is situations like this where = would have worked. My experience is that ≡ gets used here to indicate that you can replace one expression with another, typically using just syntax (what is written) rather than semantics(what is meant) – Cort Ammon Feb 09 '18 at 02:46
  • I think the usage of ≡ stems from the lower level mathematics (like Structures or Proof Theory), where they use equivalency a lot. This bubbles up to our level of math in a peculiar way where ≡ seems more "forceful" because the notation comes from those crazy math people who do things like prove that 2+2=4 using nothing but symbolic manipulation and proofs. (Botond, feel free to correct me if this isn't the "feel' you were going for) – Cort Ammon Feb 09 '18 at 02:48
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    Thank you for your recent comment. – Sebastiano Aug 20 '20 at 19:34
6

  1. There is no difference.

  2. $\ln(1)=0$.

  3. The exponential function is continuous.

Sri-Amirthan Theivendran
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