Mathematics/Calculus/Corner cases

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Derivatives

The derivative of

Question

What is ?

Solution 1

  • Let .
  • Take of both sides: .
  • Differentiate both sides: .
  • Apply the chain rule on the left-hand side: .
  • Apply the product rule on the right-hand side: .
  • Putting it together, we have .
  • Hence .

Solution 2

  • Note that , so .
  • Applying the chain rule, .
  • Applying the product rule, .
  • Therefore .

Integrals

The integral

Question

What is ?

Solution

  • We can write as .
  • Consider the series expansion of :
   .
  • We can interchange the integration and summation (we can recognize this as a special case of the Fubini/Tonelli theorems) and write
   

Limits

The limit of

Question

What is ?

Solution

  • Note that , so .
  • We can further rewrite this as .
  • As long as is continuous and the limit of exists at the point in question, the limit will commute with composition:

In our case, is continuous, so

  • The question, then, is what is .
  • As , , . In this situation we can apply l'Hôpital's rule:

  • Hence .

The limits of and

Question

What are Failed to parse (unknown function "\tendsto"): {\displaystyle \lim_{x \tendsto +\infty} x \sin \frac{1}{x}} and Failed to parse (unknown function "\tendsto"): {\displaystyle \lim_{x \tendsto -\infty} x \sin \frac{1}{x}} ?

Solution

  • Let us rewrite as .
  • As Failed to parse (unknown function "\tendsto"): {\displaystyle x \tendsto +\infty} , Failed to parse (unknown function "\tendsto"): {\displaystyle \frac{1}{x} \tendsto 0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \sin \frac{1}{x} \tendsto 0} .
  • We have ``, so we can apply l'Hôpital's rule.
  • Differentiating the numerator in , we obtain .
  • Differentiating the denominator in , we obtain .
  • Thus

Failed to parse (unknown function "\tendsto"): {\displaystyle \lim_{x \tendsto +\infty} x \sin \frac{1}{x} = \lim_{x \tendsto +\infty} \frac{\sin \frac{1}{x}}{\frac{1}{x}} = \lim_{x \tendsto +\infty} \frac{\left(\cos \frac{1}{x}\right) \left(-\frac{1}{x^2}\right)}{-\frac{1}{x^2}} = \lim_{x \tendsto +\infty} \cos \frac{1}{x} = 1. }

  • Similarly we can find that Failed to parse (unknown function "\tendsto"): {\displaystyle \lim_{x \tendsto -\infty} x \sin \frac{1}{x} = 1} .