• Two basic ways to compare the size of two quantities is to take their difference or ratio. We'll focus on the ratio since relative size is important.

• Can we find the ratio of two functions when the values of both of those functions are tending toward zero? What would this correspond to geometrically?

• Can we find the ratio of two functions when the values of both of those functions are blowing up to infinity? What would this correspond to graphically?

• Rational functions are quotients of polynomials, so naturally they compare the relative size of the polynomial in the numerator to the polynomial in the denominator using a ratio. We can also take ratios of other types of functions.

Suppose \(f(x)\) is a line of slope \(5\) through the point \((1, 0)\), and \(g(x)\) is a line of slope \(2\) through that same point.

1. Graph these functions and find formulas for them.
2. Find a formula for \(h(x) = \frac{f(x)}{g(x)}\).
3. Evaluate \(\displaystyle \lim_{x \to 1} \frac{f(x)}{g(x)}\).
4. What does the limit say about the graphs of \(f\), \(g\), and \(h\)?

Suppose \(f(x) = x^2+3x-4\) and \(g(x) = x^2-1\).

1. What is the ratio of \(f(x)\) to \(g(x)\) when \(x=1\)?
   
2. What is \({\displaystyle \lim_{x \to 1}} \frac{f(x)}{g(x)}\)?
   

Suppose \(f(x) = x^2+3x-4\) and \(g(x) = x^2-1\).

1. What is the ratio of \(f'(x)\) to \(g'(x)\) when \(x=1\)?
   
2. What is \({\displaystyle \lim_{x \to 1}} \frac{f'(x)}{g'(x)}\)?
   

Suppose \(f(x) = x^2+3x-4\) and \(g(x) = x^2-1\). Does

\[\lim_{x \to 1} \frac{x^2+3x-4}{x^2-1} \stackrel{?????}{=} \lim_{x \to 1} \frac{2x+3}{2x} = \frac{5}{2}?\]

If \(f(x) = 3x\) and \(g(x) = 2x^2+1\), what is \[\lim_{x \to \pm \infty} \frac{f(x)}{g(x)} = \lim_{x \to \pm \infty} \frac{3x}{2x^2+1}?\]

If \(f(x) = 3x^2\) and \(g(x) = 2x^2+1\), what is \[\lim_{x \to \pm \infty} \frac{f(x)}{g(x)} = \lim_{x \to \pm \infty} \frac{3x^2}{2x^2+1}?\]

If \(f(x) = 3x^3\) and \(g(x) = 2x^2+1\), what is \[\lim_{x \to \pm \infty} \frac{f(x)}{g(x)} = \lim_{x \to \pm \infty} \frac{3x^3}{2x^2+1}?\]

Suppose \(\displaystyle \frac{f(a)}{g(a)}\) is of the form \(\displaystyle \frac{0}{0}\) or \(\displaystyle \pm \frac{\infty}{\infty}\). Then \[\lim_{x \to a} \frac{f(x)}{g(x)} \stackrel{L}{=} \lim_{x \to a} \frac{f'(x)}{g'(x)}\] provided the limit exists.

• L'Hospital's rule uses a ratio \(f' / g'\) of slopes of tangent lines, not the derivative \((f/g)'\) of the quotient
• \(x \to a\) may be replaced by a one-sided limit or by \(x \to \pm \infty\)
• Use \(\stackrel{L}{=}\) to indicate L'Hospital's rule was applied

1. \(\displaystyle \lim_{x \to \pi} \frac{\sin^2(x)}{x-\pi}\)

2. \(\displaystyle \lim_{x \to \infty} \frac{e^{-x}}{x^{-1}}\)

3. \(\displaystyle \lim_{x \to 0} \frac{\sin(x)-x}{x^3}\)

4. Does \(\displaystyle y = \frac{3-5x^3}{12x^3+\pi x-1}\) have horizontal asymptotes? What are they?

5. True or false: \(\displaystyle \lim_{x\to 0} \frac{1-\cos(x)}{x^2+3x} \stackrel{L}{=} \lim_{x\to 0} \frac{\sin(x)}{2x+3} \stackrel{L}{=} \lim_{x\to 0} \frac{\cos(x)}{2} = \frac{1}{2}.\)

Indeterminate forms are expressions that do not evaluate to one unique number, such as \(\displaystyle \frac{0}{0}\) and \(\displaystyle \pm\frac{\infty}{\infty}\).

Other indeterminate forms, such as

\(\infty - \infty\)

\(0 \cdot \infty\)

\(\infty^0\)

\(0^0\)

can be converted to \(\displaystyle \frac{0}{0}\) or \(\displaystyle \pm\frac{\infty}{\infty}\) using algebra or the fact that \(e^{\ln(x)} = x\).

Use algebra to convert the forms \(\infty - \infty\) and \(0 \cdot \infty\) into either \(\displaystyle \frac{0}{0}\) or \(\displaystyle \pm\frac{\infty}{\infty}\).

1. \(\displaystyle \lim_{x \to \infty} ( \ln(2x) - \ln(3x+1) )\)

2. \(\displaystyle \lim_{x \to \pi/2} \cos(x) \sec(3x)\)

3. \(\displaystyle \lim_{t \to 0^+} t^2 \ln(t)\)

Since \(e^{ln(x)} = x\), applying \(e^{ln(\underline{~~})}\) to indeterminate powers such as limits \(\displaystyle \lim_{x\to a} f(x)^{g(x)}\) of the form \(0^0\) and \(\infty^0\) can produce a \(\displaystyle \frac{0}{0}\) or \(\displaystyle \frac{\infty}{\infty}\) form.

1. \(\displaystyle \lim_{x \to \infty} x^{1/x}\)

2. \(\displaystyle \lim_{x \to 0^+} x^{6\sin(x)}\)