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

1. A hyperbola

The first graph shows a simple hyperbola. What is the limit when ? 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 In this example, we can also say that 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.

Related differentiation topics

• Constant, Line, and Power Functions
• Exponential Functions
• Trigonometric Functions
• Constant Multiple
• Combinations: Sum, and Difference
• Combinations of Functions: Product and Quotient
• Composition of Functions (the Chain Rule)
• Transformations of Functions
• Inverses of Functions
• Hyperbolic Functions
• Linear Approximation
• Mean Value Theorem

Acknowledgements

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