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Limits at Infinity

What value does f(x) approach as x approaches infinity?

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1. A hyperbola

The first graph shows a simple hyperbola. What is the limit when c = infinity? In other words, what value does f (x) approach as x approaches infinity? Move the x slider so that x gets bigger and bigger. As you will note, f (x) approaches 0 for this example. We write this as lim x-> infinity f(x) = 0 In this example, we can also say that lim x->-infinity f(x) = 0 because the function also approaches 0 as you head towards negative infinity.

2. Asymmetric hyperbola

Select the second example. In this example, the limit at positive infinity is different from the limit at negative infinity.

3. A line

Select the third example. This is just a line. The limits at positive and negative infinity do not exist, because the function's output just keeps on getting bigger and bigger as x heads towards infinity. You can zoom out multiple times to allow you to move the slider to bigger and bigger values.

4. A sine curve

Select the fourth example. This is just a sine curve. The limits at positive and negative infinity do not exist, because the function's output keeps oscillating up and down as bigger as x heads towards infinity. You can zoom out multiple times to allow you to move the slider to bigger and bigger values.

Other differentiation topics

Acknowledgements

Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.