Properties of the graphs

The function e^{x} increases faster at infinity than any power function. Plot y = e^{x} and y = x^{3} on the same axes. Then plot y = e^{x} and y = x^{8} on the same axes. Do you still believe the statement? Make the x scale big and the y scale huge!

The function e^{−x} decreases faster at infinity than any negative power. Plot y = e^{−x} and y = x^{−2} on the same axes. Then plot y = e^{−x} and y = x^{−20} on the same axes; experiment with the scales to find the crossover point.

The function ln x increases more slowly at infinity than any positive (fractional) power. Plot y = ln x and y = x^{1/5} on the same axes. Make the x scale bigger until you find the crossover point.

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As x approaches 0, the function − ln x increases more slowly than any negative power. Plot y = − ln x and y = x^{−1/5} on the same axes. Do you believe the statement?