Limits at Infinity and Asymptotes: Learn It 5

End Behavior Cont.

Determining End Behavior for Transcendental Functions

Trigonometric functions

The six basic trigonometric functions are periodic and do not approach a finite limit as x±x±.

For example, sinxsinx oscillates between 1 and -1 (Figure 19). The tangent function xx has an infinite number of vertical asymptotes as x±x±; therefore, it does not approach a finite limit nor does it approach ±± as x±x± as shown in (Figure 20).

The function f(x) = sin x is graphed.
Figure 19. The function f(x)=sinxf(x)=sinx oscillates between 1 and -1 as x±x±
The function f(x) = tan x is graphed.
Figure 20. The function f(x)=tanxf(x)=tanx does not approach a limit and does not approach ±± as x±x±

Exponential functions

Recall that for any base b>0,b1b>0,b1, the function y=bxy=bx is an exponential function with domain (,)(,) and range (0,)(0,). If b>1,y=bxb>1,y=bx is increasing over (,)(,).If [latex]0

For the natural exponential function f(x)=exf(x)=ex, e2.718>1e2.718>1. Therefore, f(x)=exf(x)=ex is increasing on (,)(,) and the range is (0,)(0,). The exponential function f(x)=exf(x)=ex approaches as xx and approaches 0 as xx.

End behavior of the natural exponential function
xx 55 22 00 22 55
exex 0.006740.00674 0.1350.135 11 7.3897.389 148.413148.413
The function f(x) = ex is graphed.
Figure 21. The exponential function approaches zero as xx and approaches as xx.

Recall that the natural logarithm function f(x)=ln(x)f(x)=ln(x) is the inverse of the natural exponential function y=exy=ex. Therefore, the domain of f(x)=ln(x)f(x)=ln(x) is (0,)(0,) and the range is (,)(,).

The graph of f(x)=ln(x)f(x)=ln(x) is the reflection of the graph of y=exy=ex about the line y=xy=x. Therefore, ln(x)ln(x) as x0+x0+ and ln(x)ln(x) as xx.

End behavior of the natural logarithm function
xx 0.010.01 0.10.1 11 1010 100100
ln(x)ln(x) 4.6054.605 2.3032.303 00 2.3032.303 4.6054.605
The function f(x) = ln(x) is graphed.
Figure 22. The natural logarithm function approaches as xx.

Find the limits as xx and xx for f(x)=(2+3ex)(75ex)f(x)=(2+3ex)(75ex) and describe the end behavior of ff.