1.2: Graphical Investigation of Limits
- Page ID
- 203680
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- The Cartesian Coordinate System and the Quadrants
- Ordered Pairs and Graphing Relations by Point-Plotting
- Closed- and Open-Circle Notations
- Reading Values from Graphs
- An Overview of Functions
- The Domain and Range of a Function
- Graphing Functions
- Graphs of Functions
- Base Graphs
- Transformations
- Linear Functions
- Graphing Linear Equations
- Rational Functions
- Asymptotes
- Logarithmic Functions
- Graphing Logarithmic Functions
- Trigonometry
- Base Graphs of the Trigonometric Functions
Finite Limit at a Finite Number
Sketch a graph of \( f(x) = \frac{x^2 + 4x - 21}{x - 3} \).
- Does \( f(3) \) exist? If so, what is its value?
- What is the value of \( f(x) \) as \( x \to 3 \), but where \( x \neq 3 \)?
Using Graphs to Estimate Limits
Find the limit given the graph.
- Instructor Note
- Graph a piecewise function with a removable discontinuity.
Revisiting the Existence of a Limit
Evaluate \[ \lim_{x \to 0} \sin\left( \dfrac{\pi}{x} \right). \nonumber \]
Revisiting One-Sided Limits
Let's take a look at a piecewise function.
Revisiting Infinite Limits at Finite Numbers
Evaluate each of the following limits, if possible, using graphs.
- \(\displaystyle \lim_{x \to 3^−} \frac{1}{(x - 3)^2}\)
- \(\displaystyle \lim_{x \to 3^+} \frac{1}{(x - 3)^2}\)
- \( \displaystyle \lim_{x \to 3}\frac{1}{(x - 3)^2}\)
The function \(f(x)\) is said to have a vertical asymptote at \(x = a\) if any of the following are true.\[\lim_{x \to a^−}f(x)=+\infty \nonumber \]\[\lim_{x \to a^−}f(x)=-\infty \nonumber \]\[\lim_{x \to a^+}f(x)=+\infty \nonumber \]\[\lim_{x \to a^+}f(x)=-\infty \nonumber \]\[\lim_{x \to a}f(x)=+\infty \nonumber \]\[\lim_{x \to a}f(x)=-\infty \nonumber \]
Conceptual Investigation of an Infinite Limit
\[ \dfrac{\to \text{finite nonzero number}}{\to 0} \qquad \text{and} \qquad \dfrac{\to 0}{\to 0} \nonumber \]If a limit has the form\[ \dfrac{\to \text{finite nonzero number}}{\to 0}, \nonumber \]we can guarantee the limit is a form of infinity. We use the signs of the numerator and denominator to investigate whether the limit is \( +\infty \) or \( -\infty \).
Another limit we will run into has the form\[ \dfrac{\to \text{finite number}}{\to \pm \infty}. \nonumber \]In this case, the limit will become \( 0 \).
Let \( f(x) = \frac{N(x)}{D(x)} \), where \( \displaystyle \lim_{x \to a} N(x) \) is a finite, nonzero constant, and \( D(x) \) is approaching 0 as \( x \) approaches \( a \). Then\[ \lim_{x \to a} \dfrac{N(x)}{D(x)} \nonumber \]is either \( \infty \) or \( -\infty \). The sign is determined by comparing the signs of the numerator and denominator of the ratio as \(x\) is approaching \(a\).
Evaluate each of the following limits.
- \( \displaystyle \lim_{x \to -7}{ \frac{2}{x + 7} } \)
- \( \displaystyle \lim_{x \to -7}{ \frac{2}{(x + 7)^2} } \)
- \( \displaystyle \lim_{x \to -5^-}{ \ln{(x^2 - 25)} } \)
Revisiting Evaluation of Limits from Graphs
Let's finish this section by revisiting the graphical investigation of limits.
Let \(f(x)\) be the function graphed below. Use this Interactive Element to answer the questions.
Interact: Move the point to help investigate limits.
Questions: Use the graph to estimate the value of some limits.
Some Warnings
Tables are the absolute worst way to evaluate a limit.
Graphs are the second-to-worst method to evaluate a limit.
If a denominator within a limit is tending to 0, this does not mean the limit is "undefined." Here's a hint: undefined is not used in calculus!