Understanding Limits via Graphs and Tables
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Input and Output Values of $f(x) = 3x + 1$ as $x$ Approaches 2. $x$ approaches 2 from the left. $x$ approaches 2 from the right. A table showing x values of 1, 1.5, 1.9, 1.99, 1.999, 2.001, 2.01, 2.1, 2.5, 3 and corresponding $f(x)$ values of 4, 5.5, 6.7, 6.97, 6.997, 7.003, 7.03, 7.3, 8.5, 10. $y$ approaches 7 from both directions.
This question includes visual content: The image contains three Cartesian coordinate graphs labeled f(x), g(x), and h(x), each showing a linear function passing through (0, 1) and (2, 7), with specific visual indicators for the limit at x=2. Below these is a table titled 'Input and Output Values of f(x) = 3x + 1 as x Approaches 2', listing x-values approaching 2 from the left (1, 1.5, 1.9, 1.99, 1.999) and from the right (2.001, 2.01, 2.1, 2.5, 3), with corresponding f(x) values that show y approaching 7.
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Step by Step Written Solution
Hi Ömer, let's explore the concept of a limit by looking at this function and its table of values.
Understanding Limits
We are examining the function f of x equals three x plus one as x approaches two.
First, let's look at the left side of our table. As x approaches two from numbers smaller than two, like one point nine nine nine, the output values get closer and closer to seven.
Approach from the Left
| x | f(x) |
|---|---|
| 1 | 4 |
| 1.5 | 5.5 |
| 1.9 | 6.7 |
| 1.99 | 6.97 |
| 1.999 | 6.997 |
Notice how the output y values are climbing up towards seven as we get nearer to x equals two.
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