Lesson 10 Limits and Continuity

10.1 Lecture Content

Video: Numerical and Graphical Approaches to Limits

Video: Limits and Continuity

10.2 Lecture Notes

10.2.1 Estimate a Limit Using a Table of Values

To estimate a limit using a table:

  • Choose values of \(x\) that get closer to the target from both sides (left and right).
  • Observe how the function values \(f(x)\) behave.
  • If the values approach a single number, that is the estimated limit.

Example: If \(f(x) = \frac{x^2 - 1}{x - 1}\), use values close to 1 (e.g., 0.9, 0.99, 1.01, 1.1) to estimate \(\lim_{x \to 1} f(x)\).

10.2.2 Use Factoring to Evaluate Limits

Factoring helps eliminate indeterminate forms like \(\frac{0}{0}\).

Steps:

  1. Factor the numerator and denominator.
  2. Cancel common factors.
  3. Substitute the limit value.

Example: Evaluate \(\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\): \[ \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \Rightarrow \lim_{x \to 2} = 4 \]

10.2.3 One-Sided Limits and Vertical Asymptotes

  • Left-hand limit: \(\lim_{x \to a^-} f(x)\)
  • Right-hand limit: \(\lim_{x \to a^+} f(x)\)

If the function approaches infinity or negative infinity near a point, it may indicate a vertical asymptote.

Example: \[ \lim_{x \to 0^-} \frac{1}{x} = -\infty,\quad \lim_{x \to 0^+} \frac{1}{x} = \infty \]

10.2.4 Dominant Term Analysis for Limits at Infinity

To evaluate limits as \(x \to \infty\), focus on the dominant (highest degree) terms.

Steps:

  1. Identify the highest power term in the numerator and denominator.
  2. Divide all terms by the dominant power of \(x\).
  3. Simplify and take the limit.

Example: \[ \lim_{x \to \infty} \frac{3x^2 + 5}{x^2 - 2x} = \lim_{x \to \infty} \frac{3 + 5/x^2}{1 - 2/x} = \frac{3}{1} = 3 \]

10.2.5 Continuity of Piecewise Functions

To determine if a piecewise function is continuous at a point:

  1. Check function is defined: \(f(a)\) exists.
  2. Evaluate both one-sided limits: \(\lim_{x \to a^-} f(x)\) and \(\lim_{x \to a^+} f(x)\).
  3. Check limit equals value: \(\lim_{x \to a} f(x) = f(a)\)

If all three conditions are met, the function is continuous at \(x = a\).

Example:

\[ f(x) = \begin{cases} x^2, & x < 2 \\ 4x - 4, & x \geq 2 \end{cases} \]

Check continuity at \(x = 2\): - \(f(2) = 4\) - \(\lim_{x \to 2^-} = 4\), \(\lim_{x \to 2^+} = 4\) - Since limits and value agree: continuous

10.3 Examples

10.3.1 Limits

Video: Solutions

  1. Find the following limit:

\[ \lim_{x \to 7} \frac{x^3 -3x^2 - 29x +7}{x - 7} \]

  1. Estimate the limit numerically or state that the limit does not exist:

\[ \lim_{x \to 121} \frac{\sqrt{x} - 11}{x - 121} \]

  1. Use numerical or graphical evidence to determine the left and right-hand limits of the function.

\[ \lim_{x \to 4^-} \frac{|5x - 20|}{x-4} \]

\[ \lim_{x \to 4^+} \frac{|5x - 20|}{x-4} \]

10.3.2 One Sided Limits

Video: Solutions

  1. A function \(f(x)\) is said to have a jump discontinuity at \(x = a\) if:

    1. \(\displaystyle \lim_{x \to a^-} f(x)\) exists.
    2. \(\displaystyle \lim_{x \to a^+} f(x)\) exists.
    3. The left and right limits are not equal.

Let

\[ f(x) = \begin{cases} 6x-6, & \text{if } x < 8, \\ \frac{2}{x+6}, & \text{if } x \ge 8 \end{cases} \]

Show that \(f(x)\) has a jump discontinuity at \(x = 8\).

  1. For what value of the constant \(c\) is the function \(f\) continuous on \((-\infty, \infty)\) where

\[ f(x) = \begin{cases} x^2 - c & \text{if } t < 6, \\ cx + 9, & \text{if } t \ge 6 \end{cases} \]

Find \(c\).

  1. Given the function below, determine if the function is continuous at the point \(x = -4\). If not, indicate why.

\[ f(x) = \begin{cases} 2x+9, & \text{if } x < -4, \\ -2x - 7, & \text{if } x > -4 \end{cases} \]

  1. Given the function below, determine if the function is continuous at the point \(x = -1\). If not, indicate why.

\[ f(x) = \ln \|x+1\| \]

10.3.3 One Sided Limits

Video: Solutions

  1. Let

\[ f(x) = \begin{cases} -\frac{12}{x+6}, & \text{if } x \le 0 \\ \frac{14}{x-7}, & \text{if } x > 0 \end{cases} \]

Compute the quantities below. a. \[ \lim_{x \to -3^-} f(x) = \quad \] b. \[ \lim_{x \to -3^+} f(x) = \quad \] c.
\[ f(-3) = \quad \]

  1. Evaluate the limit: \[ \lim_{x \to \infty} \frac{5 + 5x}{8 - 4x} \]

  2. Evaluate the limit:

\[ \lim_{x \to \infty} \frac{8x + 7}{6x^2 - 7x + 7} \]

  1. Evaluate the limit:

\[ \lim_{x \to \infty} \frac{\sqrt{9 + 2x^2}}{7 + 6x} \]

  1. Evaluate:

\[ \lim_{t \to \infty} \frac{-3t - 9}{\sqrt{t^2 -4t + 5}} \]

  1. Evaluate the following limits:

    1. \[ \lim_{x \to \infty}\frac{8}{e^x - 9} \]

    2. \[ \lim_{x \to -\infty} \frac{8}{e^x - 9} \]

10.4 Practice Problems

Practice the techniques discussed in class and in the online videos by solving the following examples.

  1. Estimate \(\lim_{x \to 2} e^{x-2}\).

  2. Estimate \(\lim_{x \to 0^+} \frac{-2}{x^2}\).

  3. Evaluate \(\lim_{x \to 0} \frac{x-3}{x-1}\).

  4. Evaluate \(\lim_{x \to 2} \frac{x^{2} - 8x + 12}{x-2}\).

  5. Evaluate \(\lim_{x \to -2} \frac{x+2}{x^{2} - 4}\).

  6. Evaluate \(\lim_{x \to \infty} e^{-x}\).

  7. Evaluate \(\lim_{x \to \infty} \frac{3x - x^{6} + 2}{3x^{3} + 2x}\).

  8. Evaluate \(\lim_{x \to -\infty} \frac{1 - 3x}{2x^{2} + 3}\).

  9. Find a number \(b\) so that \[ f(x) = \begin{cases} 5x - 6 & x \leq 2 \\ -3x + b & x > 2 \end{cases} \] is continuous everywhere.

  10. Find a number \(a\) so that \[ f(x) = \begin{cases} ax - 3 & x \leq 3 \\ x + a & x > 3 \end{cases} \] is continuous everywhere.

  11. (Applied) The percentage of movie advertising as a share of newspapers’ total advertising revenue from 1995 to 2004 can be approximated by \[ p(t) = \begin{cases} -0.07t + 6 & t \leq 4 \\ 0.3t + 17 & t > 4 \end{cases} \] where \(t\) is the time since 1995.

  1. Compute \(\lim_{t \to 4^-} p(t)\) and \(\lim_{t \to 4^+} p(t)\) and interpret each answer.

  2. Is the function \(p\) continuous at \(t = 4\)? What does the answer tell you about newspaper revenues?

10.5 Self Assessment

Time yourself and try to solve the following questions within twenty minutes.

  1. Evaluate \(\lim_{x \to -1} \frac{4x^2 + 1}{x}\).

  2. Evaluate \(\lim_{x \to -4} \frac{x + 4}{x^{2} - 16}\).

  3. Evaluate \(\lim_{x \to \infty} \frac{60 + e^{-x}}{2 - e^{-x}}\).

  4. Your friend Fiona claims that the study of limits is silly; all you ever need to do to find the limit as \(x\) approaches \(a\) is substitute \(x = a\). Give two examples that show she is wrong.

  5. Find a number \(b\) so that \[ f(x) = \begin{cases} 2x + 1 & x \leq -3 \\ -x + b & x > -3 \end{cases} \] is continuous everywhere.

10.6 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill D CON COM
Estimate a limit using a table of values or by direct substitution.
Use factoring to evaluate a limit.
Use dominant terms analysis to evaluate limits approaching infinity.
Find a parameter to make a piecewise function continuous.
Solve applied problems involving limits.