Chapter 1: Limits

Section 1.5: Limits at Infinity and Infinite Limits

Example 1.5.8

$$ Graph the function $f\left(x\right)\=\arctan \left(x\right)$, and determine all its asymptotes. $$ Solution $$ Figure 1.5.8(a) suggests that there are only horizontal asymptotes for the graph of $\arctan \left(x\right)$. In fact, the limits at $\pm \infty$ support this observation.   $\underset{x\to -\infty }{lim}\arctan \left(x\right)$ = $-\frac{1}{2}\mathrm{\π}$$$ $$$$ $\underset{x\to \infty }{lim}\arctan \left(x\right)$ = $\frac{1}{2}\mathrm{\π}$$$       >  p1:=plot(arctan(x),x=-5..5,tickmarks=[spacing(1,0),spacing(Pi/2,0)],view=[-5..5,-2..2],color=black): p2:=plot([Pi/2,-Pi/2],x=-5..5,color=red,linestyle=dot): plots:-display(p1,p2,scaling=constrained); >    Figure 1.5.8(a)   Graph of $\arctan \left(x\right)$ (black) and its horizontal asymptotes (red) $$

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