Solved Examples: Estimating Limits from Tables with Detailed Explanations

📚 Quick Study Guide

• 🔢 Definition of a Limit: The limit of a function $f(x)$ as $x$ approaches $c$ is $L$, written as $\lim_{x \to c} f(x) = L$, if $f(x)$ gets arbitrarily close to $L$ as $x$ gets arbitrarily close to $c$ (but not necessarily equal to $c$).
• 📊 Estimating Limits from Tables: To estimate a limit from a table, examine the values of $f(x)$ as $x$ approaches $c$ from both the left (values less than $c$) and the right (values greater than $c$).
• 📈 One-Sided Limits:
  • 🔍 The limit from the left: $\lim_{x \to c^-} f(x)$
  • 🧭 The limit from the right: $\lim_{x \to c^+} f(x)$
• ✔️ Existence of a Limit: The limit $\lim_{x \to c} f(x)$ exists and is equal to $L$ if and only if both one-sided limits exist and are equal to $L$. That is, $\lim_{x \to c^-} f(x) = L$ and $\lim_{x \to c^+} f(x) = L$.
• 🚫 Non-Existence of a Limit: A limit does not exist if:
  • 💥 The one-sided limits are not equal.
  • ♾️ The function increases or decreases without bound as $x$ approaches $c$.
  • oscillatory behavior near $c$.

Practice Quiz

1. Question 1: Consider the table below. What is the estimated value of $\lim_{x \to 2} f(x)$?

   x	1.9	1.99	1.999	2.001	2.01	2.1
   f(x)	3.8	3.98	3.998	4.002	4.02	4.2

   1. A) 2
   2. B) 3
   3. C) 4
   4. D) Does Not Exist
2. Question 2: Given the table below, estimate $\lim_{x \to 0} g(x)$.

   x	-0.1	-0.01	-0.001	0.001	0.01	0.1
   g(x)	1.9	1.99	1.999	2.001	2.01	2.1

   1. A) 0
   2. B) 1
   3. C) 2
   4. D) Does Not Exist
3. Question 3: Use the table to estimate $\lim_{x \to 3} h(x)$.

   x	2.9	2.99	2.999	3.001	3.01	3.1
   h(x)	8.9	8.99	8.999	9.001	9.01	9.1

   1. A) 3
   2. B) 6
   3. C) 9
   4. D) Does Not Exist
4. Question 4: Estimate $\lim_{x \to 1} f(x)$ from the table.

   x	0.9	0.99	0.999	1.001	1.01	1.1
   f(x)	-0.1	-0.01	-0.001	0.001	0.01	0.1

   1. A) -1
   2. B) 0
   3. C) 1
   4. D) Does Not Exist
5. Question 5: Consider the table and estimate $\lim_{x \to 5} p(x)$.

   x	4.9	4.99	4.999	5.001	5.01	5.1
   p(x)	24.9	24.99	24.999	25.001	25.01	25.1

   1. A) 5
   2. B) 10
   3. C) 20
   4. D) 25
6. Question 6: From the table, estimate $\lim_{x \to -2} q(x)$.

   x	-2.1	-2.01	-2.001	-1.999	-1.99	-1.9
   q(x)	-4.1	-4.01	-4.001	-3.999	-3.99	-3.9

   1. A) -4
   2. B) -2
   3. C) 0
   4. D) Does Not Exist
7. Question 7: Given the values in the table, what is $\lim_{x \to 4} r(x)$?

   x	3.9	3.99	3.999	4.001	4.01	4.1
   r(x)	15.9	15.99	15.999	16.001	16.01	16.1

   1. A) 4
   2. B) 8
   3. C) 16
   4. D) Does Not Exist