3.4.1: Resources and Key Concepts — Read

From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

Resources

Indeterminate Forms: Expressions arising from limits whose value cannot be determined by direct substitution or simple arithmetic of limits. This section focuses on:
- \(\frac{0}{0}\): An indeterminate form where both the numerator and denominator of a quotient approach zero.
- \(\frac{\infty}{\infty}\): An indeterminate form where both the numerator and denominator of a quotient approach infinity (or negative infinity).
- \(0 \cdot \infty\): An indeterminate form arising from the product of a term approaching zero and a term approaching infinity.
- \(\infty - \infty\): An indeterminate form arising from the difference of two terms both approaching infinity.
- \(0^0, \infty^0, 1^\infty\): Indeterminate forms involving exponents.
Growth Rates of Functions:
- Function \(g(x)\) grows more rapidly than \(f(x)\) as \(x \to \infty\) if \( \displaystyle \lim_{x \to \infty} \frac{g(x)}{f(x)} = \infty\) or, equivalently, \(\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)} = 0\).
- Functions \(f(x)\) and \(g(x)\) grow at the same rate as \(x \to \infty\) if there exists a constant \(M \neq 0\) such that \(\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)} = M\).

Theorem: L'Hôpital's Rule (\(\frac{0}{0}\) Case): Suppose \(f\) and \(g\) are differentiable functions over an open interval containing \(a\), except possibly at \(a\). If \(\displaystyle \lim_{x \to a} f(x) = 0\) and \(\displaystyle \lim_{x \to a} g(x) = 0\), then \(\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \displaystyle \lim_{x \to a} \frac{f^{\prime}(x)}{g^{\prime}(x)}\), assuming the limit on the right exists or is \(\infty\) or \(-\infty\). This result also holds for one-sided limits, or if \(a=\infty\) or \(a=-\infty\).
Theorem: L'Hôpital's Rule (\(\frac{\infty}{\infty}\) Case): Suppose \(f\) and \(g\) are differentiable functions over an open interval containing \(a\), except possibly at \(a\). Suppose \(\displaystyle \lim_{x \to a} f(x) = \infty\) (or \(-\infty\)) and \(\displaystyle \lim_{x \to a} g(x) = \infty\) (or \(-\infty\)). Then, \(\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \displaystyle \lim_{x \to a} \frac{f^{\prime}(x)}{g^{\prime}(x)}\), assuming the limit on the right exists or is \(\infty\) or \(-\infty\). This result also holds if the limit is infinite, if \(a=\infty\) or \(a=-\infty\), or if the limit is one-sided.

Abusive Notation (\(\frac{0}{0}\) or \(\frac{\infty}{\infty}\)): The notation \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) does not mean actual division by zero or division of infinities. It represents a quotient of limits, where each limit individually evaluates to 0 or \(\infty\).
Applying L'Hôpital's Rule Incorrectly:
- Applying the rule when the limit is not an indeterminate form of type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).
- Differentiating the quotient \(\frac{f(x)}{g(x)}\) using the Quotient Rule instead of taking the quotient of the derivatives, \(\frac{f^{\prime}(x)}{g^{\prime}(x)}\).
Not Converting Other Indeterminate Forms: Forgetting to algebraically manipulate indeterminate forms like \(0 \cdot \infty\), \(\infty - \infty\), \(0^0\), \(\infty^0\), or \(1^\infty\) into a \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) form before attempting to apply L'Hôpital's Rule.
Errors in Algebraic Manipulation: Mistakes made when rewriting expressions to fit the \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) form, or when simplifying after applying L'Hôpital's Rule.
Stopping Too Early or Applying Too Many Times: After applying L'Hôpital's Rule, one must re-evaluate the new limit. If it's still indeterminate and the conditions are met, the rule can be applied again. If it's determinate, the rule should not be applied further.

3.4.1: Resources and Key Concepts is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.