michaelbailey2005
michaelbailey2005 7d ago โ€ข 0 views

Solved Examples: Estimating Limits from Tables with Detailed Explanations

Hey there! ๐Ÿ‘‹ Ever struggled with estimating limits from tables? It can seem tricky, but with a few solved examples and clear explanations, you'll be a pro in no time! Let's dive in! ๐Ÿงฎ
๐Ÿงฎ Mathematics

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blakesmith1989 Jan 2, 2026

๐Ÿ“š Quick Study Guide

  • ๐Ÿ”ข 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 $c$).
  • ๐Ÿ“Š 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 $c$).
  • ๐Ÿ“ˆ One-Sided Limits:
    • ๐Ÿ” The limit from the left: $\lim_{x \to c^-} f(x)$
    • ๐Ÿงญ The limit from the right: $\lim_{x \to c^+} f(x)$
  • โœ”๏ธ 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 $\lim_{x \to c^+} f(x) = L$.
  • ๐Ÿšซ Non-Existence of a Limit: A limit does not exist if:
    • ๐Ÿ’ฅ The one-sided limits are not equal.
    • โ™พ๏ธ 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 $\lim_{x \to 2} f(x)$?
    x1.91.991.9992.0012.012.1
    f(x)3.83.983.9984.0024.024.2
    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-0.1-0.01-0.0010.0010.010.1
    g(x)1.91.991.9992.0012.012.1
    1. A) 0
    2. B) 1
    3. C) 2
    4. D) Does Not Exist
  3. Question 3: Use the table to estimate $\lim_{x \to 3} h(x)$.
    x2.92.992.9993.0013.013.1
    h(x)8.98.998.9999.0019.019.1
    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.
    x0.90.990.9991.0011.011.1
    f(x)-0.1-0.01-0.0010.0010.010.1
    1. A) -1
    2. B) 0
    3. C) 1
    4. D) Does Not Exist
  5. Question 5: Consider the table and estimate $\lim_{x \to 5} p(x)$.
    x4.94.994.9995.0015.015.1
    p(x)24.924.9924.99925.00125.0125.1
    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 $\lim_{x \to 4} r(x)$?
    x3.93.993.9994.0014.014.1
    r(x)15.915.9915.99916.00116.0116.1
    1. A) 4
    2. B) 8
    3. C) 16
    4. D) Does Not Exist
Click to see Answers
  1. Answer: C) 4
  2. Answer: C) 2
  3. Answer: C) 9
  4. Answer: B) 0
  5. Answer: D) 25
  6. Answer: A) -4
  7. Answer: C) 16

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