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).


Exponential functions
Recall that for any base b>0,b≠1b>0,b≠1, 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, e≈2.718>1e≈2.718>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 x→∞x→∞ and approaches 0 as x→−∞x→−∞.
xx | −5−5 | −2−2 | 00 | 22 | 55 |
exex | 0.006740.00674 | 0.1350.135 | 11 | 7.3897.389 | 148.413148.413 |

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 x→0+x→0+ and ln(x)→∞ln(x)→∞ as x→∞x→∞.
xx | 0.010.01 | 0.10.1 | 11 | 1010 | 100100 |
ln(x)ln(x) | −4.605−4.605 | −2.303−2.303 | 00 | 2.3032.303 | 4.6054.605 |

Find the limits as x→∞x→∞ and x→−∞x→−∞ for f(x)=(2+3ex)(7−5ex)f(x)=(2+3ex)(7−5ex) and describe the end behavior of ff.