Taming the Indeterminate

The Problem: Indeterminate Forms

In calculus, we are often interested in finding limits. Sometimes, when we try to evaluate a limit by simple substitution, we get forms that are meaningless or “indeterminate.” They don’t tell us what the limit actually is.

The most common indeterminate forms are:

1. $$\frac{0}{0}$$​

2. $$\frac{\infty}{\infty}$$​

Other forms like $$0\cdot\infty$$, $$\inf-\infty$$, $$0^{0}$$, $$1^{\infty}$$, and
$$\inf^{0}$$ can also be indeterminate, but they often require a bit of algebra to be massaged into the $$\frac{0}{0}$$ or $$\frac{\infty}{\infty}$$​ forms.

• Why are they indeterminate? Consider $$$$\frac{0}{0}​. It could be hiding any number.

  • $$\lim_{x\to0}\frac{2x}{x}$$ (The limit is 2)

  • $$\lim_{x\to0}\frac{5x}{x}$$ (The limit is 5)

  • $$\lim_{x\to0}\frac{x}{x^{2}}$$ (The limit is infinite)

The form $$\frac{0}{0}$$​ is a question, not an answer. L’Hôpital’s Rule provides a method to find the answer.

The Solution: L’Hôpital’s Rule

Statement of the Rule:
Suppose f and g are differentiable functions and $$g(x)\neq0$$ on an open interval containing a (except possibly at a itself).

If

$$\$\$\backslash lim\_\{x\backslash to\; a\}\backslash frac\{f(x)\}\{g(x)\}\$\$\; produces\; the\; indeterminate\; form\; \$\$\backslash frac\{0\}\{0\}\$\$\; or\; \$\$\backslash frac\{\backslash infty\}\{\backslash infty\}\$\$$$$$,$$

then

$$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f^\prime(x)}{g^\prime(x)}$$​,

provided the limit on the right exists or is infinite.

In simple terms: When you get $$\$\$\backslash frac\{0\}\{0\}\$\$$$​ or $$\frac{\infty}{\infty}$$, you can take the derivative of the top and the derivative of the bottom separately, and then take the limit again.

Step-by-Step Application & Examples

Step 1: Always verify that the limit leads to $$\$\$\backslash frac\{0\}\{0\}\$\$$$​​ or $$\frac{\infty}{\infty}$$​ by direct substitution.
Step 2: Apply L’Hôpital’s Rule by differentiating the numerator and denominator separately.
Step 3: Evaluate the new limit. If it is a determinate value, you are done. If the new limit is also $$\$\$\backslash frac\{0\}\{0\}\$\$$$​ or $$\frac{\infty}{\infty}$$​, apply L’Hôpital’s Rule again.

Example 1: The Classic $$\$\$\backslash frac\{0\}\{0\}\$\$$$​
$$\lim_{x\to 0}\frac{\sin(x)}{x}$$

1. Check: $$\frac{\sin(0)}{0}=\frac{0}{0}$$ ✅

2. Apply L’Hôpital’s Rule: Differentiate top and bottom.

   • Derivative of numerator: $$\frac{d}{dx}(\sin(x))=\cos(x)$$

   • Derivative of denominator: $$\frac{d}{dx}(x)=1$$

3. The new limit is: $$\lim_{x\to 0}\frac{\cos(x)}{1}=\frac{1}{1}=1$$

∴ $$\lim_{x\to 0}\frac{\sin(x)}{x}=1$$

Example 2: Multiple Applications
$$\lim_{x\to\inf}\frac{e^{x}}{x^{2}}$$

1. Check: As x approach to infinity then$$e^{x}\longrightarrow\inf , x^{2}\longrightarrow\inf$$.so,$$\frac{\infty}{\infty}$$​ ✅

2. Apply L’Hôpital’s Rule:

   • Derivative of numerator: $$\frac{d}{dx}e^{x}=e^{x}$$

   • Derivative of denominator: $$\frac{d}{dx}x^{2}=2x$$

   • New limit: $$\lim_{x\to\inf}\frac{e^{x}}{2x}$$​

3. Check again: $$\frac{\infty}{\infty}$$​ ✅ Apply L’Hôpital’s Rule again.

   • Derivative of numerator: $$\frac{d}{dx}e^{x}=e^{x}$$

   • Derivative of denominator: $$\frac{d}{dx}2x=2$$

   • New limit: $$\lim_{x\to\infty}\frac{e^{x}}{2}$$

∴ $$\lim_{x\to\infty}\frac{e^{x}}{2}=\infty$$

Example 3: A $$\frac{\infty}{\infty}$$​ that goes to 0
$$\lim_{x\to\infty}\frac{\ln(x)}{x}$$

1. Check: $$\frac{\ln(\infty)}{\infty}=\frac{\infty}{\infty}$$​ ✅

2. Apply L’Hôpital’s Rule:

   • Derivative of numerator: $$\frac{d}{dx}\ln(x)=\frac{1}{x}$$​

   • Derivative of denominator: $$\frac{d}{dx}(x)=1$$

   • New limit: $$\lim_{x\to\infty}\frac{\frac{1}{x}}{1}=\lim_{x\to\infty}\frac{1}{x}=0$$

∴ $$\lim_{x\to\infty}\frac{\ln(x)}{x}=0$$
(This shows that the function x grows faster than ln⁡x as x→∞.)

Handling Other Indeterminate Forms

L’Hôpital’s Rule directly applies only to $$\frac{0}{0}$$​ and $$\frac{\infty}{\infty}$$​. For other forms, we must use algebra to rewrite them.

Case 1: $$0\cdot\infty$$

• Strategy: Rewrite the product as a quotient.

• Example: Find $$\lim_{x\to0^{+}}x\ln(x)$$.

  • This is 0⋅(−∞). We rewrite it as $$\lim_{x\to0^{+}}\frac{\ln(x)}{1/x}$$

  • Now it’s $$\frac{-\infty}{infty}$$​ ✅. Apply L’Hôpital’s Rule.

  • $$\lim_{x\to0^{+}}\frac{\ln(x)}{1/x}=\lim_{x\to0^{+}}\frac{1/x}{-1/x^{2}}=\lim_{x\to0^{+}}\frac{1}{x}\cdot\frac{-x^{2}}{1}=\lim_{x\to0^{+}}(-x)=0$$

Case 2: $$\infty-\infty$$

• Strategy: Combine the terms into a single fraction.

• Example: Find$$\lim_{x\to0}\left(\frac{1}{\sin x}-\frac{1}{x}\right)$$

  • This is $$\infty-\infty$$. Combine: $$\lim_{x\to0}\frac{x-\sin(x)}{x\sin(x)}$$

  • Now it’s $$\frac{0}{0}​$$ ✅. Apply L’Hôpital’s Rule (you’ll likely need to apply it multiple times here). The result is 0.

Case 3: $$0^{0},\infty^{0},1^{\infty}$$

• Strategy: Use the exponential log trick. Set $y=[\text{expression}]$, then take the natural log of both sides:$$\ln(y)=\ln(limit)$$. This will bring the exponent down. The new limit will often be a $$0\cdot\infty$$ form, which we can convert to $$\frac{0}{0}$$​ or $$\frac{\infty}{\infty}$$​

• After finding the limit for $$\ln(y)$$, don’t forget to exponentiate to solve for y.

V. Important Warnings and Caveats

1. IT ONLY APPLIES TO $$\frac{0}{0}$$​ or $$\frac{\infty}{\infty}$$! This is the most common mistake. If you apply it to a form like $$\frac{1}{0}$$, you will get the wrong answer. Always check the form first.

2. Differentiate Top and Bottom Separately! Do not use the Quotient Rule. You are taking the derivative of the numerator function and the derivative of the denominator function independently.

3. Know When to Stop: Sometimes, applying L’Hôpital’s Rule leads to a loop. If you find yourself going in circles, there might be a better algebraic way to solve the limit.

4. It’s a Tool, Not a Magic Wand: For some limits, basic algebra (like factoring and canceling, as we saw with rational functions) is simpler and more efficient. Always see if simplification is possible first.

VI. Why Does L’Hôpital’s Rule Work? (The Intuition)

Imagine two functions f(x) and g(x) that are both 0 at x=a. We want to know the behavior of their ratio $$\frac{f(x)}{g(x)}$$​ near a.

Their derivatives, f′(a) and g′(a), tell us their instantaneous rates of change at a. If we zoom in incredibly close to the point (a,0), the graphs of f and g look like their tangent lines.

• Near a, f(x)≈f′(a)(x−a)

• Near a, g(x)≈g′(a)(x−a)

Therefore, near $a$:

$$\frac{f(x)}{g(x)}\approx\frac{f^\prime(a)(x-a)}{g^\prime(a)(x-a)}=\frac{f^\prime(a)}{g^\prime(a)}$$​

The (x−a) factors cancel, and we are left with the ratio of the derivatives. L’Hôpital’s Rule makes this intuitive idea precise.

Summary

L’Hôpital’s Rule is an indispensable technique for evaluating limits that result in the indeterminate forms $$\frac{0}{0}$$​​ and $$\frac{\infty}{\infty}$$​. It often simplifies seemingly intractable problems by leveraging the power of derivatives.

The Mantra: “Zero over zero or infinity over infinity? Take the derivative of the top and the derivative of the bottom and try again.”

Master this rule, and a whole new class of limit problems will become manageable, paving the way for your work with series and improper integrals later in calculus.