Selecting Procedures for Determining Limits - AP Calculus BC Study Guide

The Core Idea: Selecting Procedures for Determining Limits

Determining the limit of a function is not a monolithic process; it requires a strategic approach based on the structure of the function and its behavior near a specific point. The core idea of this topic is to develop a systematic decision-making framework for evaluating limits. The first step is always direct substitution. The result of this initial step—whether it's a defined real number, an expression indicating an infinite limit, or an indeterminate form like $\frac{0}{0}$—dictates the subsequent procedure.

This topic organizes the primary analytical techniques for finding limits. When direct substitution fails to produce a clear answer, you must select the appropriate algebraic tool, such as factoring and canceling for rational functions, or using the conjugate for functions involving radicals. For more complex functions, such as those defined by inequalities, a different approach like the Squeeze Theorem is required. The central skill is to recognize the form of the limit and match it to the correct, most efficient analytical method.

Key Theorems and Procedures

The evaluation of limits relies on a set of specific analytical methods. The choice of method is determined by the form of the function and the result of direct substitution.

The Four Main Analytical Procedures

1. Direct Substitution: For a function $f(x)$ that is continuous at $x=c$, the limit can be found by substituting $c$ directly into the function.

   $$\underset{x\to c}{\lim }f(x)=f(c)$$

   This is always the first method to attempt.

2. Factoring and Canceling: This technique is used when direct substitution results in the indeterminate form $\frac{0}{0}$, particularly with rational functions. The goal is to factor the numerator and denominator to find a common factor that can be canceled, thereby removing the discontinuity at $x=c$.

   • Form:${\lim }_{x\to c}\frac{p(x)}{q(x)}$ where $p(c)=0$ and $q(c)=0$.

   • Procedure: Factor $p(x)$ and $q(x)$, cancel the common factor $(x-c)$, and re-evaluate the limit of the simplified function.

3. Using the Conjugate: This method is applied when direct substitution yields $\frac{0}{0}$ and the expression involves a square root, typically in the form $\sqrt{a}-\sqrt{b}$ or $\sqrt{a}-b$. Multiplying the numerator and denominator by the conjugate of the expression containing the radical often resolves the indeterminacy.

   • Form:${\lim }_{x\to c}\frac{\sqrt{f(x)}-L}{g(x)}$ where substitution yields $\frac{0}{0}$.

   • Procedure: Multiply the numerator and denominator by the conjugate, $\sqrt{f(x)}+L$. Simplify the expression, which often leads to a cancellation, and then re-evaluate the limit.

4. Using One-Sided Limits: This approach is essential for determining the existence of a two-sided limit, especially for piecewise-defined functions, functions with absolute values, or at points of potential discontinuity. The two-sided limit ${\lim }_{x\to c}f(x)$ exists if and only if the left-hand and right-hand limits are equal.

   $$\underset{x\to c}{\lim }f(x)=L⟺\underset{x\to c^{-}}{\lim }f(x)=L\text{ and }\underset{x\to c^{+}}{\lim }f(x)=L$$

The Squeeze Theorem

The Squeeze Theorem is a powerful tool for finding the limit of a function that is "squeezed" between two other functions.

Theorem Statement:

Let $g(x)$, $f(x)$, and $h(x)$ be functions satisfying the inequality $g(x)\le f(x)\le h(x)$ for all $x$ in some open interval containing $c$, except possibly at $x=c$ itself. If

$$\underset{x\to c}{\lim }g(x)=\underset{x\to c}{\lim }h(x)=L$$

then

$$\underset{x\to c}{\lim }f(x)=L$$

Understanding Indeterminate Forms

A critical concept in selecting a limit procedure is the recognition of an indeterminate form. When direct substitution into a limit expression results in $\frac{0}{0}$, this does not mean the limit is undefined or equal to zero. Instead, it signals that the limit's value cannot be determined from this form alone and that further analysis is required.

The form $\frac{0}{0}$ implies that both the numerator and the denominator are approaching zero. The true value of the limit depends on the relative rates at which they approach zero. The algebraic techniques of factoring and using the conjugate are designed to manipulate the function into an equivalent form (for $x\ne c$) where this indeterminacy is resolved.

For example, in ${\lim }_{x\to 2}\frac{x^2-4}{x-2}$, direct substitution gives $\frac{0}{0}$. This indicates a "hole" in the graph of the function at $x=2$. By factoring and canceling, $\frac{(x-2)(x+2)}{x-2}=x+2$ (for $x\ne 2$), we are essentially "patching" the hole. The limit of the new, simplified function as $x\to 2$ gives us the y-value of that hole, which is the value the original function approaches. Therefore, understanding that $\frac{0}{0}$ is a signal to begin an analytical procedure, not to stop, is fundamental.

Core Concepts & Rules

• Always Start with Direct Substitution: The first step in evaluating any limit, ${\lim }_{x\to c}f(x)$, is to calculate $f(c)$. If this yields a real number, that number is the limit.

• Recognize Indeterminate Forms: If direct substitution results in $\frac{0}{0}$, the limit is indeterminate. This requires you to use an alternative algebraic method to find the limit.

• Match the Method to the Function:

  • For rational functions resulting in $\frac{0}{0}$, use the factoring and canceling method.

  • For functions with radicals resulting in $\frac{0}{0}$, use the conjugate method.

• Check Both Sides for Certain Functions: For piecewise functions or functions with absolute values, you must evaluate the one-sided limits from the left ($x\to c^{-}$) and the right ($x\to c^{+}$). The two-sided limit exists only if these one-sided limits are equal.

• Use the Squeeze Theorem for Bounded Functions: If a function $f(x)$ is trapped between two other functions, $g(x)$ and $h(x)$, use the Squeeze Theorem. To apply it, you must show that the limits of the outer functions ($g(x)$ and $h(x)$) are equal at the point in question.

Step-by-Step Example 1: Using the Conjugate

Problem: Determine the limit ${\lim }_{x\to 4}\frac{\sqrt{x}-2}{x-4}$.

Step 1: Attempt Direct Substitution

Substitute $x=4$ into the expression:

$$\frac{\sqrt{4}-2}{4-4}=\frac{2-2}{4-4}=\frac{0}{0}$$

This is an indeterminate form, so we must use an alternative procedure.

Step 2: Select the Appropriate Procedure

The expression contains a square root in the numerator and resulted in $\frac{0}{0}$. This indicates that multiplying by the conjugate is the correct method. The conjugate of $\sqrt{x}-2$ is $\sqrt{x}+2$.

Step 3: Apply the Procedure

Multiply the numerator and the denominator by the conjugate:

$$\underset{x\to 4}{\lim }\frac{\sqrt{x}-2}{x-4}\cdot \frac{\sqrt{x}+2}{\sqrt{x}+2}$$

Simplify the numerator. Recall that $(a-b)(a+b)=a^2-b^2$.

$$\underset{x\to 4}{\lim }\frac{(\sqrt{x}{)}^2-2^2}{(x-4)(\sqrt{x}+2)}=\underset{x\to 4}{\lim }\frac{x-4}{(x-4)(\sqrt{x}+2)}$$

Cancel the common factor of $(x-4)$:

$$\underset{x\to 4}{\lim }\frac{1}{\sqrt{x}+2}$$

Step 4: Re-evaluate the Limit by Direct Substitution

Now, substitute $x=4$ into the simplified expression:

$$\frac{1}{\sqrt{4}+2}=\frac{1}{2+2}=\frac{1}{4}$$

Step 5: State the Conclusion

The limit is $\frac{1}{4}$.

$$\underset{x\to 4}{\lim }\frac{\sqrt{x}-2}{x-4}=\frac{1}{4}$$

Step-by-Step Example 2: Exam-Style Application (Squeeze Theorem)

Problem: A function $f(x)$ satisfies the inequality $4\cos (\pi x)\le f(x)\le x^2+3$ for all $x$ in the interval $(0,2)$. Determine ${\lim }_{x\to 1}f(x)$. Justify your answer.

Step 1: Identify the Structure and Select the Procedure

The problem provides an inequality that bounds the function $f(x)$ between two other functions. This is the classic setup for applying the Squeeze Theorem.

Step 2: Define the Bounding Functions

Let the lower bounding function be $g(x)=4\cos (\pi x)$.

Let the upper bounding function be $h(x)=x^2+3$.

We are given $g(x)\le f(x)\le h(x)$.

Step 3: Evaluate the Limits of the Bounding Functions

We need to find the limits of $g(x)$ and $h(x)$ as $x\to 1$. We can use direct substitution for both.

• Limit of the lower bound:

  $$\underset{x\to 1}{\lim }g(x)=\underset{x\to 1}{\lim }4\cos (\pi x)=4\cos (\pi \cdot 1)=4(-1)=-4$$

• Limit of the upper bound:

  $$\underset{x\to 1}{\lim }h(x)=\underset{x\to 1}{\lim }(x^2+3)=(1{)}^2+3=1+3=4$$

Step 4: Analyze the Results and Draw a Conclusion

The limit of the lower bound is -4, and the limit of the upper bound is 4. Since ${\lim }_{x\to 1}g(x)\ne {\lim }_{x\to 1}h(x)$, the conditions for the Squeeze Theorem are not met. Therefore, the Squeeze Theorem cannot be used to determine ${\lim }_{x\to 1}f(x)$ from the information given.

Note: This is a common exam-style problem that tests whether you check the conditions of the theorem. If the upper bound had been, for example, $h(x)=-x^3-3$, then ${\lim }_{x\to 1}h(x)=-4$, and the conclusion would be that ${\lim }_{x\to 1}f(x)=-4$ by the Squeeze Theorem.

Using Your Calculator

The methods described in this topic—factoring, using the conjugate, and applying the Squeeze Theorem—are analytical. On the AP Exam, you must show the algebraic steps to receive full credit. A calculator cannot perform these symbolic manipulations for you.

However, a graphing calculator is an excellent tool for verifying your answer or building intuition about a limit.

To check a limit ${\lim }_{x\to c}f(x)$:

1. Graphical Method:

   • Graph the function $Y_1=f(x)$.

   • Use the TRACE function. Enter values very close to $c$ from both the left (e.g., $c-0.001$) and the right (e.g., $c+0.001$).

   • Observe if the y-values are approaching the same number. This number is your suspected limit. Be mindful of the viewing window, as a poor window can be misleading.

2. Numerical (Table) Method:

   • Go to TBLSET (Table Setup). Set TblStart to $c$ and ΔTbl (delta table) to a small number like $0.001$.

   • Go to TABLE. The table will display the y-values for x-values just above and below $c$.

   • Observe the pattern of the y-values as $x$ gets closer to $c$. If they converge to a single value, this confirms your analytical result.

For the Squeeze Theorem example, you could graph $Y_1=4\cos (\pi x)$ and $Y_2=x^2+3$ to visually confirm that they do not approach the same value at $x=1$.

AP Exam Quick Hit

Common Question Types

• Indeterminate Form Requiring Algebra: You will be given a limit that evaluates to $\frac{0}{0}$ via direct substitution. You must show the correct algebraic manipulation (factoring or conjugate) to find the limit.

  • Example: Find ${\lim }_{x\to -3}\frac{x^2+x-6}{x+3}$.

• Squeeze Theorem Justification: You will be given a function bounded by two other functions and asked to find its limit. A full justification requires stating the limits of the outer functions and concluding with a reference to the Squeeze Theorem.

  • Example: If $2-x^2\le g(x)\le 2\cos (x)$ for $x\ne 0$, find ${\lim }_{x\to 0}g(x)$.

• One-Sided Limits from a Piecewise Function: You will be given a piecewise function and asked to find a one-sided or two-sided limit at the point where the function definition changes.

  • Example: For $f(x)=\{\begin{array}{ll}{x}^2+1& x<2\\ 3x-1& x\ge 2\end{array}$, find ${\lim }_{x\to 2}f(x)$.

Common Mistakes

• Stopping at $\frac{0}{0}$: A very common error is to conclude that a limit is undefined or does not exist simply because direct substitution yields $\frac{0}{0}$. This form is indeterminate and is a signal to do more work.

• Incorrect Conjugate Multiplication: When using the conjugate method, students sometimes forget to multiply both the numerator and the denominator by the conjugate, or they make an algebraic error when expanding the terms (e.g., incorrectly squaring the radical term).

• Assuming the Squeeze Theorem Applies: Students may see an inequality and immediately assume the Squeeze Theorem works. It is crucial to calculate the limits of both bounding functions. If they are not equal, the theorem cannot be used to determine the limit.

• Errors in Factoring: Simple algebraic mistakes when factoring polynomials (e.g., difference of squares, sum/difference of cubes, or trinomials) will lead to an incorrect simplified function and an incorrect limit.

• Confusing One-Sided and Two-Sided Limits: Forgetting that for a two-sided limit to exist, the left-hand and right-hand limits must be calculated and must be equal. This is especially critical for piecewise functions at the boundary points.