Choose an exponential function. As an example, choose f(x) = y^x
Calculate the limit of the function as x approaches infinity. Using the example function:
For 0 < y < 1, limit (y^x) = 0. This is because as you increase x to infinity any fraction between 0 and 1 will trend to 0. For example:
If y is .40 and x is 3, f(x) = 0.064. If y is 0.40 and x is 12, f(x) = 0.000017. If y is 0.95 and x is 5, f(x) = 0.77. If y is 0.95 and x is 15, f(x) = 0.46. If y is 0.95 and x is 60, f(x) = 0.046. In each case, f(x) trends toward 0, which means the limit is 0.
For y = 1, limit (y^x) = 1. 1 to any power is 1
For y > 1, limit (y^x) = infinity. As x increases on any number greater than 1, the value of f(x) increases continuously to infinity.
Calculate the limit of the function as x approaches negative infinity. Using the example function:
First, we have to adjust the function to account for a negative x or "-x": f(x) = y^-x = (1/y^x)
For 0 < y < 1, limit [(1/y^x)] = infinity. Here's why:
If y = 0.40 and x = 5, f(x) = 1/[0.40^5] = 1/ 0.01024 = 97.65. This number will increase as x increases.
For y > 1, limit [(1/y^x)] = 0. Here's why.
If y = 4 and x = 5, f(x) = 1/(4^5) = 1/1024 = 0.000975. As x increases, the denominator will continue to increase and f(x) will continue to decrease and trend toward zero