Calculus 2, Fall 2016

Prelude to L'Hospital's rule

• 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.

Example 1: When f and g tend to zero

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 $$h$$?

Example 2: When f and g tend to zero

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 $${\displaystyle \lim_{x \to 1}} \frac{f(x)}{g(x)}$$?

Example 2: When f and g tend to zero

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 $${\displaystyle \lim_{x \to 1}} \frac{f'(x)}{g'(x)}$$?

Example 2: When f and g tend to zero

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

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

Example 3: When f and g blow up to $$\infty$$

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

Example 4: When f and g blow up to $$\infty$$

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

Example 5: When f and g blow up to $$\infty$$

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

L'Hospital's rule

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

Examples

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

Other indeterminate forms

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 $$e^{\ln(x)} = x$$.

Examples

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

Indeterminate powers

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