Top Banner
1.2 Evaluating Limits and One-Sided Limits
17

Theorem 3 One-Sided and Two-Sided Limits · 2018. 9. 1. · Evaluating Limits Algebraically B. lim 𝑥→2 𝑥2+2𝑥+4 𝑥+2 Example 1: Determine the limits algebraically.

Jan 01, 2021

Download

Documents

Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
  • 1.2 Evaluating Limits

    and One-Sided Limits

  • Evaluating Limits

    We have already studied the three ways to evaluate limits:

    Graphically

    Numerically (with a table)

    Algebraically

    Today we will explore algebraically evaluating limits a little more

  • Evaluating Limits Algebraically

    A. lim𝑥→3

    [𝑥2 2 − 𝑥 ]

    Example 1: Determine the limits algebraically.

  • Evaluating Limits Algebraically

    B. lim𝑥→2

    𝑥2+2𝑥+4

    𝑥+2

    Example 1: Determine the limits algebraically.

  • Evaluating Limits Algebraically

    A. lim𝑡→2

    𝑡2−3𝑡+2

    𝑡2−4

    Example 2: Determine the limits algebraically.

  • Evaluating Limits Algebraically

    B. lim𝑥→0

    1

    2+𝑥−1

    2

    𝑥

    Example 2: Determine the limits algebraically.

  • Evaluating Limits AlgebraicallyExample 2: Determine the limits algebraically.

    C. lim𝑥→9

    𝑥−3

    𝑥−9

  • Evaluating Limits Algebraically

    You need to “adjust” a function when the original substitution gives

    you 0

    0; this is called indeterminate form

    Note that something like 2

    0or −

    5

    0are different; use graphs or tables to

    determine those limits

    “Adjusting” can mean expanding and simplifying, factoring and

    cancelling, combining multiple fractions into one, or multiplying by

    the conjugate

  • One-Sided and Two-Sided Limits

    For a limit to exist, the function must approach the same

    value from both sides. Or,

    lim lim limx c x c x c

    f x f x f x

  • Example 3

    A. Find

    B. Does the

    exist?

    2

    limx

    f x

    2

    limx

    f x

    x

    y

  • Example 4

    Given c = 2,

    A. Draw the graph of f.

    B. Determine and

    C. Does exist? If so, what is it? If not, explain.

    3 , 2

    1, 22

    x x

    f x xx

    limx c

    f x

    lim .x c

    f x

    limx c

    f x

    x

    y

  • Evaluating Limits, Example 5

  • Non-existing Limits

    There are 3 reasons for a non-existing limit

    1. The left and right limits do not match (or one does not exist)

    2. Unbounded behavior (aka an asymptote)

    3. Oscillating behavior

  • Non-existing Limits, Example 6

    Algebraically, how do we know that these do not have limits?

  • Non-existing Limits, Example 7

    First, look at the graph or a table for the functions.

    Algebraically, how do we know that these do not have limits?

    A. B. Exception**

    https://sites.math.washington.edu/~conroy/general/sin1overx/

    0

    1limsinx x

    0

    1lim sinx

    xx

    https://sites.math.washington.edu/~conroy/general/sin1overx/

  • In Summary…

    There are 3 ways to find a limit:

    Numerically (a table)

    Graphically

    Algebraically (substitution, adjusting)

    In order for a limit to exist, the left and right limits must be the same

    The function does not have to exist at the c value, nor does the

    function value have to equal the limit in order for the limit to exist

    There are 3 times a limit will not exist

    One-sided limits are not equal/do not exist

    Unbounded behavior

    Oscillating behavior

  • Homework!

    P. 66 #15 – 27 odd, 37, 39, 49, 51, 63