4.3: L'Hospital's Rule: Comparing Rates of Growth

Consider Problem 3c on Homework 4: Find \[\lim_{h\rightarrow 0}\dfrac{(4 + h^2) - 16}{h}\]

In using Limit Laws, we get $0/0$. This is called an indeterminate form.

Previously, we had to preprocess the problem before using limit laws. We will now use the derivative to deal with four indeterminate forms for limits: \[\dfrac{0}{0} \qquad \dfrac{\infty}{\infty} \qquad 0 \cdot \infty \qquad \infty - \infty \]

Indeterminate Quotients

Suppose $\displaystyle \lim_{x\rightarrow a} \dfrac{f(x)}{g(x)} = \dfrac{0}{0}$. This situation is pictured below: Replacing $f(x)$ with $f'(x)$ and $g(x)$ with $g'(x)$ near $x = a$ allows us to look at slopes instead.

L'Hospital's Rule
Suppose $f$ and $g$ are differentiable and $g'(x) \neq 0$ near $x = a$. Suppose either $\displaystyle\lim_{x\rightarrow a} f(x) = \lim_{x\rightarrow a} g(x) = 0$ or $\displaystyle\lim_{x\rightarrow a} f(x) = \lim_{x\rightarrow a} g(x) = \pm\infty$.
Then \[\lim_{x\rightarrow a} \dfrac{f(x)}{g(x)} = \lim_{x\rightarrow a} \dfrac{ f'(x)}{g'(x)}\] if the limit on the right side exists or is $\pm \infty$.
Find the following limits:
  1. $\displaystyle \lim_{x\rightarrow 1} \dfrac{\ln x}{x - 1}$
  2. $\displaystyle \lim_{x\rightarrow \infty} \dfrac{e^x}{x^2}$
  3. $\displaystyle \lim_{x\rightarrow \infty} \dfrac{\ln x}{\sqrt{x}}$
  4. $\displaystyle \lim_{x\rightarrow \pi^-} \dfrac{\sin x}{1 - \cos x}$

Which Functions Grow Fastest?

Suppose $\displaystyle\lim_{x\rightarrow \infty}f(x) = \infty$ and $\displaystyle\lim_{x\rightarrow \infty} g(x) = \infty$.

We say $f(x)$ approaches infinity more quickly than $g(x)$ if \[\lim_{x\rightarrow \infty} \dfrac{f(x)}{g(x)} = \infty\] and $f(x)$ approaches infinity more slowly than $g(x)$ if \[\lim_{x\rightarrow \infty} \dfrac{f(x)}{g(x)} = 0\]

Rank the following functions in order of how quickly they approach infinity: \[y = \ln x \qquad y = x \qquad y = e^{0.1 x} \qquad y = \sqrt{x}\]
Analyze the limiting behavior of \[f(x) = \dfrac{x + x^4}{1 + x^2 + e^x}\] by considering the dominant terms in the numerator and denominator.

Indeterminate Products

Let's now deal with the indeterminate form $0 \cdot \infty$.

Let's say $\displaystyle \lim_{x\rightarrow a} f(x) = 0$ and $\displaystyle \lim_{x\rightarrow a} g(x) = \infty$.

Evaluate $\displaystyle\lim_{x\rightarrow 0^+} x \cdot \ln x$.
Evaluate $\displaystyle\lim_{x\rightarrow -\infty} x^2\cdot e^x$.
We modeled the blood alcohol concentration after one alcoholic drink with the function \[C(t) = 0.0225t e^{-0.0467t}\] Find $\displaystyle \lim_{t \rightarrow \infty} C(t)$ and discuss if it makes sense or not.

Indeterminate Differences

To deal with the indeterminate form $\infty - \infty$, try to put it into $0/0$ or $\infty/\infty$ indeterminate form then use L'Hospital's Rule.

You can also try to get it into $\infty \cdot \infty$ form, which is $\infty$, not indeterminate.

Compute \[\lim_{x\rightarrow \frac{\pi}{2}^-} (\sec x - \tan x)\]
Compute \[\lim_{x\rightarrow \infty} (e^x - x)\]