Selecting Procedures for Determining Limits - AP Calculus AB Study Guide

The Core Idea: Selecting Procedures for Determining Limits

Determining the limit of a function is a foundational skill in calculus, but it is not a monolithic process. The central idea of this topic is that a diverse toolkit of methods is required to evaluate limits, and the key to success lies in selecting the appropriate procedure for a given function and point. Depending on how the function is presented—as an equation, a graph, or a table of data—and the initial result of direct substitution, a specific strategy must be employed.

The process is often hierarchical. One might begin with direct substitution, and if that fails to produce a determinate value, other analytical techniques like algebraic manipulation or L'Hôpital's Rule become necessary. For functions that are difficult to handle algebraically, graphical analysis, numerical estimation from tables, or theoretical tools like the Squeeze Theorem provide alternative pathways. This topic synthesizes various limit-finding techniques into a cohesive problem-solving framework, emphasizing the strategic thinking required to navigate different scenarios, from simple continuous functions to more complex indeterminate forms and end-behavior analysis.

Key Theorems and Procedures

The determination of limits relies on a collection of methods and two key theorems for specific situations. The choice of method depends on the form of the function and the result of an initial attempt at direct substitution.

Analytical Procedures

1. Direct Substitution: For continuous functions, the limit can be found by substituting the value $c$ into $f(x)$.

2. Algebraic Manipulation: Used when direct substitution results in an indeterminate form like $\frac{0}{0}$. Techniques include:

   • Factoring and canceling common factors.

   • Multiplying by the conjugate.

   • Finding a common denominator for complex fractions.

3. One-Sided Limits: The limit ${\lim }_{x\to c}f(x)$ exists if and only if the left-hand limit and the right-hand limit are equal: ${\lim }_{x\to c^{-}}f(x)={\lim }_{x\to c^{+}}f(x)$. This is crucial for piecewise functions and at vertical asymptotes.

4. End Behavior Analysis: Used to determine limits as $x\to \infty$ or $x\to -\infty$. For rational functions, this often involves comparing the degrees of the numerator and denominator.

Key Theorems

1. The Squeeze Theorem:

   If $h(x)\le f(x)\le g(x)$ for all $x$ in an open interval containing $c$, except possibly at $c$ itself, and if

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

   then ${\lim }_{x\to c}f(x)=L$. This theorem is useful for finding limits of functions that are "squeezed" between two other functions with a common limit.

2. L'Hôpital's Rule:

   If ${\lim }_{x\to c}f(x)=0$ and ${\lim }_{x\to c}g(x)=0$, or if ${\lim }_{x\to c}f(x)=\pm \infty$ and ${\lim }_{x\to c}g(x)=\pm \infty$, then

   $$\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\underset{x\to c}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$$

   provided the limit on the right side exists or is $\pm \infty$. This rule applies only to indeterminate forms of type `\frac{0}{0}$and$\frac{\infty}{\infty}. ## Understanding the Hierarchy of Methods The most critical skill in evaluating limits is knowing which method to apply and when. There is a logical progression to follow when faced with a limit problem. 1. **Start with Direct Substitution:** Always attempt to plug the value $c into the function $f(x)$ first.

   • If you get a real number $L$, then ${\lim }_{x\to c}f(x)=L$. The process is complete.

   • If you get a non-zero number divided by zero (e.g., $\frac{k}{0}$ where $k\ne 0$), this indicates a vertical asymptote. The limit will be $\infty$, $-\infty$, or it will not exist. You must then investigate the one-sided limits by testing values slightly less than and slightly greater than $c$.

   • If you get an indeterminate form ($\frac{0}{0}$ or $\frac{\infty }{\infty }$), you must do more work. This is a signal to proceed to the next step.

3. Address Indeterminate Forms: When you have $\frac{0}{0}$ or $\frac{\infty }{\infty }$, you have two primary options:

   • Algebraic Manipulation: Try to rewrite the function so that the factor causing the zero in the denominator can be canceled. This is the standard approach for rational functions and functions involving radicals.

   • L'Hôpital's Rule: If the function is a ratio of two differentiable functions and the conditions are met, this is often the most direct method, especially for functions involving transcendental components (e.g., trigonometric, exponential, logarithmic).

4. Special Cases:

   • The Squeeze Theorem: This is a more specialized tool. Look for its use when you see a trigonometric function like $\sin (x)$ or $\cos (x)$ (which are bounded between -1 and 1) multiplied by a function that approaches 0, such as $x^2\sin (\frac{1}{x})$ as $x\to 0$.

   • Limits at Infinity: To evaluate ${\lim }_{x\to \pm \infty }f(x)$, analyze the end behavior. For rational functions, compare the degrees of the numerator and denominator. For other functions, consider the dominant terms as $x$ becomes very large.

5. Graphical and Numerical Approaches:

   • If you are given a graph, observe the $y$-value the function approaches as $x$ gets closer to $c$ from both the left and the right.

   • If you are given a table, observe the trend in the $y$-values as the $x$-values get progressively closer to $c$.

Core Concepts & Rules

• A limit can be determined from a graph by observing the $y$-value a function approaches as $x$ approaches a given point from both sides.

• A limit can be estimated from a table of data by analyzing the function's output values as the input values get closer to a specific point.

• Direct substitution is the primary method for evaluating limits and should always be attempted first.

• If direct substitution yields an indeterminate form ($\frac{0}{0}$ or $\frac{\infty }{\infty }$), further analysis using algebraic manipulation or L'Hôpital's Rule is required.

• L'Hôpital's Rule is a powerful technique for indeterminate forms that involves taking the derivatives of the numerator and denominator separately.

• The Squeeze Theorem allows for the determination of a limit for a function by comparing it to two other functions with known, equal limits.

• The existence of a two-sided limit, ${\lim }_{x\to c}f(x)$, is contingent upon the left-hand and right-hand limits being equal: ${\lim }_{x\to c^{-}}f(x)={\lim }_{x\to c^{+}}f(x)$.

• Limits involving infinity ($x\to \pm \infty$) describe the end behavior of a function and correspond to horizontal asymptotes.

Step-by-Step Example 1: Algebraic Manipulation

Problem: Find the limit ${\lim }_{x\to 3}\frac{x^2-x-6}{x-3}$.

Step 1: Attempt Direct Substitution

Substitute $x=3$ into the expression:

$$\frac{(3{)}^2-(3)-6}{3-3}=\frac{9-3-6}{0}=\frac{0}{0}$$

This is an indeterminate form, which signals that we must use another method.

Step 2: Select an Appropriate Procedure

Since the expression is a rational function, algebraic manipulation is a suitable method. We will try to factor the numerator to see if a common factor with the denominator exists.

Step 3: Apply the Algebraic Procedure

Factor the quadratic in the numerator: $x^2-x-6=(x-3)(x+2)$.

Now, substitute the factored form back into the limit expression:

$$\underset{x\to 3}{\lim }\frac{(x-3)(x+2)}{x-3}$$

Step 4: Simplify the Expression

For $x\ne 3$, we can cancel the $(x-3)$ term from the numerator and denominator:

$$\underset{x\to 3}{\lim }(x+2)$$

Step 5: Evaluate the Final Limit

Now, use direct substitution on the simplified expression:

$$(3)+2=5$$

Therefore, ${\lim }_{x\to 3}\frac{x^2-x-6}{x-3}=5$.

Step-by-Step Example 2: Exam-Style Application (L'Hôpital's Rule)

Problem: Find the limit ${\lim }_{x\to 0}\frac{1-\cos (x)}{x^2}$.

Step 1: Attempt Direct Substitution

Substitute $x=0$ into the expression:

$$\frac{1-\cos (0)}{0^2}=\frac{1-1}{0}=\frac{0}{0}$$

This is an indeterminate form.

Step 2: Select an Appropriate Procedure

The expression is a ratio of differentiable functions, and we have the indeterminate form $\frac{0}{0}$. This meets the conditions for L'Hôpital's Rule.

Step 3: Apply L'Hôpital's Rule

Take the derivative of the numerator and the derivative of the denominator separately.

• Derivative of numerator: $\frac{d}{dx}(1-\cos (x))=\sin (x)$

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

The new limit to evaluate is:

$$\underset{x\to 0}{\lim }\frac{\sin (x)}{2x}$$

Step 4: Evaluate the New Limit

Attempt direct substitution on the new expression:

$$\frac{\sin (0)}{2(0)}=\frac{0}{0}$$

We have another indeterminate form. Since the conditions are still met, we can apply L'Hôpital's Rule a second time.

Step 5: Apply L'Hôpital's Rule Again

Take the derivative of the new numerator and denominator:

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

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

The new limit is:

$$\underset{x\to 0}{\lim }\frac{\cos (x)}{2}$$

Step 6: Evaluate the Final Limit

Use direct substitution on the final expression:

$$\frac{\cos (0)}{2}=\frac{1}{2}$$

Therefore, ${\lim }_{x\to 0}\frac{1-\cos (x)}{x^2}=\frac{1}{2}$.

Using Your Calculator

A graphing calculator is a powerful tool for estimating limits or for checking an analytical answer. It is primarily used for the graphical and numerical methods mentioned in the Essential Knowledge.

Estimating a Limit from a Table

To estimate ${\lim }_{x\to c}f(x)$:

1. Enter the function $f(x)$ into Y1=.

2. Go to TBLSET (Table Setup).

3. Set TblStart to $c$, the value $x$ is approaching.

4. Set ΔTbl (Delta Table) to a very small number, such as $0.001$.

5. Go to TABLE. The table will display $y$-values for $x$-values very close to $c$. Observe the trend in the Y1 column as $x$ approaches $c$ from both above and below. This provides a numerical estimate of the limit.

Estimating a Limit from a Graph

1. Enter the function $f(x)$ into Y1=.

2. Use ZOOM to find an appropriate viewing window around the $x$-value $c$.

3. Use the TRACE function. Enter `x$-values that are very close to $c$ from both the left ($c-0.001$) and the right ($c+0.001$).

4. Observe the corresponding $y$-values. If they are approaching the same number, that is your graphical estimate for the limit.

Important Note: For free-response questions, an answer supported only by a calculator's graph or table is generally not sufficient. You must show the analytical steps (e.g., algebra, L'Hôpital's Rule). Use the calculator to confirm your result.

AP Exam Quick Hit

Common Question Types

• Limit from a Piecewise Function Graph: You will be shown a graph with jumps, holes, and/or cusps and asked to determine one-sided and two-sided limits at the points of discontinuity. For example, "Use the graph of $f(x)$ above to find ${\lim }_{x\to 2^{-}}f(x)$ and ${\lim }_{x\to 2}f(x)$."

• Indeterminate Form Requiring L'Hôpital's Rule: A multiple-choice question will present a limit that evaluates to $\frac{0}{0}$ or $\frac{\infty }{\infty }$, often involving trigonometric or exponential functions. For example, "What is ${\lim }_{x\to 0}\frac{e^{3x}-1}{\sin (2x)}$?"

• Limit at Infinity (Horizontal Asymptote): You will be asked to find ${\lim }_{x\to \infty }f(x)$ for a rational function, which requires comparing the degrees of the numerator and denominator. For example, "Find the value of ${\lim }_{x\to \infty }\frac{3x^2-5x+1}{7-2x^2}$."

Common Mistakes

• Applying L'Hôpital's Rule Incorrectly: Students often apply the quotient rule to $\frac{f(x)}{g(x)}$ instead of taking the derivatives of $f(x)$ and $g(x)$ separately. Remember, it is $\frac{f^{\prime }(x)}{g^{\prime }(x)}$, not $(\frac{f}{g}{)}^{\prime }(x)$.

• Applying L'Hôpital's Rule to a Determinate Form: A common trap is to use L'Hôpital's Rule when direct substitution yields a valid number (e.g., $\frac{0}{5}=0$). The rule only applies to indeterminate forms. Always check with direct substitution first.

• Confusing the Limit with the Function Value: The limit as $x\to c$ describes what $f(x)$ approaches, which may not be the same as $f(c)$. This is especially true at a hole in a graph, where the limit exists but the function value may be different or undefined.

• Algebraic Errors: Simple mistakes in factoring, canceling terms, or working with conjugates can lead to an incorrect answer even when the correct procedure is chosen.

• Stating $\frac{0}{0}$ as the Answer: The expression $\frac{0}{0}$ is an indeterminate form that indicates more work is needed. It is never the final answer to a limit problem.