#AP Calculus AB/BC: Estimating Limits from Graphs 📈

Hey there, future calculus champ! Let's get you prepped to ace those limit questions on the AP exam. This guide is designed to be your go-to resource, especially the night before the test. We'll make sure everything clicks, and you feel confident and ready! Let's dive in!

#🎯 Understanding the Concept of Limits

Before we jump into graphs, let's quickly review what limits are all about. Think of a limit as the value a function is approaching, not necessarily the value it is at a specific point.

To quickly review how to define limits, check out Key Topic 1.2!

#One-Sided Limits ↔️

Sometimes, we need to look at how a function behaves as we approach a point from either the left or the right. These are called one-sided limits. Think of it like approaching a destination from different directions.

• Left-Hand Limit: Approaching from the left side: $$\lim_{{x \to a^-}} f(x)$$
• Right-Hand Limit: Approaching from the right side: $$\lim_{{x \to a^+}} f(x)$$

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#📊 Estimating Limits from Graphs

Graphs are super helpful for visualizing how a function behaves. Let's see how we can use them to estimate limits.

#🧙‍♂️ Tips and Tricks

#🚧 Challenges with Graphical Estimation

Graphs are great, but they're not perfect. Here are some things to watch out for:

• Issues of Scale: Graphs can be misleading if the scale is not chosen carefully. Always check the axes!
• Missed Function Behavior: Sometimes, graphs might miss important details, like sudden jumps or oscillations. Be cautious!

#❌ When Limits Might Not Exist

It's important to know when a limit doesn't exist. Here are the most common scenarios:

#Limit DNE: Unbounded Function

If the function goes to infinity (or negative infinity) as x approaches a certain value, the limit does not exist. Think of vertical asymptotes!

Graph with vertical asymptote at x = 1.

Image Courtesy of Study.com

#Limit DNE: Oscillating Function

If the function oscillates wildly as x approaches a certain value, the limit does not exist. Think of functions like sin(1/x) near x=0.

Graph of sin(1/x) with oscillations.

Image Courtesy of Study.com

#Limit DNE: Left and Right Limits Differ

If the left-hand limit and the right-hand limit are different, the overall limit does not exist. This usually happens at a jump discontinuity.

Graph of a discontinuity at x = 1 between the left and right limits.

Image Courtesy of Study.com

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#📝 Evaluating Limits: Practice Problems

Alright, let's put your skills to the test! Use the graph below to evaluate the following limits:

Graph of piecewise function f(x).

Image Courtesy of Mathwarehouse.com

1. $$\lim_{x\to -2}f(x)$$
2. $$\lim_{x\to -3}f(x)$$
3. $$\lim_{x\to 0}f(x)$$
4. $$\lim_{x\to 2}f(x)$$

#✏️ Practice Problems: Answers

1. $$\lim_{x\to -2}f(x) = DNE$$
   • $$\lim_{x\to -2^-}f(x) = -1$$, because as f(x) approaches 2 from the left side, the limit is -1. - $$\lim_{x\to -2^+}f(x) = -3$$, because as f(x) approaches 2 from the right side, the limit is -3. - Different limits from each side mean that the limit at the point, which is -2, doesn’t exist.
2. $$\lim_{x\to -3}f(x) = 4$$
3. $$\lim_{x\to 0}f(x) = 0$$
4. $$\lim_{x\to 2}f(x) = 1$$

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#💫 Conclusion

You've got this! Remember to visualize the point, trace along the graph, and check for one-sided limits. Be aware of the limitations of graphical representations and the situations in which limits might not exist. With these insights, you're well-prepared to tackle this aspect of calculus with confidence.

Keep practicing, and you'll master the art of estimating limits from graphs. Good luck! ☘️

Practice Question

Multiple Choice Questions:

1. The graph of a function f is shown below. Which of the following statements is true?

   

   (A) $$\lim_{x \to 1} f(x) = 2$$ (B) $$\lim_{x \to 1} f(x) = 1$$ (C) $$\lim_{x \to 1} f(x)$$ does not exist (D) $$f(1) = 2$$

2. Given the function g(x), which of the following statements is true about the limit at x=a where there is a jump discontinuity?

   (A) The limit exists and is equal to the value of the function at x=a (B) The limit exists and is equal to the average of the left and right limits (C) The limit does not exist because the left and right limits are different (D) The limit exists and is equal to the left side limit.

Free Response Question:

Consider the function h(x) defined by the graph below:

(a) Find $$\lim_{x \to -3} h(x)$$ (b) Find $$\lim_{x \to -2^-} h(x)$$ (c) Find $$\lim_{x \to -2^+} h(x)$$ (d) Does $$\lim_{x \to -2} h(x)$$ exist? Justify your answer. (e) Find $$h(0)$$

Answer Key and Scoring Rubric:

Multiple Choice Answers:

1. (C) The limit does not exist because the left-hand and right-hand limits are different.
2. (C) The limit does not exist because the left and right limits are different.

Free Response Scoring:

(a) $$\lim_{x \to -3} h(x) = 4$$ (1 point) (b) $$\lim_{x \to -2^-} h(x) = -1$$ (1 point) (c) $$\lim_{x \to -2^+} h(x) = -3$$ (1 point) (d) No, $$\lim_{x \to -2} h(x)$$ does not exist because the left-hand limit (-1) is not equal to the right-hand limit (-3). (2 points: 1 for no, 1 for correct justification) (e) $$h(0) = 0$$ (1 point)