Chapter 1: Limits
Section 1.6: Continuity
Where is the function Fx=tan(4−x3/4+4⁢x2−2⁢x+3x2+1) continuous?
Because of the term x3/4, the interval −∞,0 is excluded from the domain of F.
The graph of F in Figure 1.6.5(a) shows the vertical asymptote whose equation is x=x^≐0.6400732558 and the horizontal asymptote y=tan4≐1.157821282.
Thus, F is continuous on −∞,x^⋃x^,∞.
G := (abs(4-x^(3/4)) + 4*x^2-2*x+3)/(x^2+1):
F := tan(G):
Figure 1.6.5(a) Graph of Fx for x≥0
Recognize Fx as the composition Fx=tangx, where gx=4−x3/4+4⁢x2−2⁢x+3x2+1
Control-drag (or type) gx=…
Context Panel: Assign Function
gx=4−x3/4+4⁢x2−2⁢x+3x2+1→assign as functiong
Find the horizontal asymptote: y=limx→∞Fx
Expression palette: Limit operator
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻10
limx→∞tangx = tan⁡4→at 10 digits1.157821282
Find the vertical asymptote
Figure 1.6.5(b) is a graph of gx along with the lines y=k π/2,k=1,3,5. The vertical asymptotes for tanx occur at the odd half-multiples of π, so Figure 1.6.5(b) is drawn to see where gx, the argument of tangx, assumes one of these values. The only one is 3 π/2.
It is also useful to know that g0 = 7, and that the minimum of gx is approximately 3.45, found by applying the Optimization≻Minimize option from the Context Panel:
gx = −4+x3/4+4⁢x2−2⁢x+3x2+1→minimize3.44835673863100434,x=2.36115843528260
G:=(abs(4-x^(3/4)) + 4*x^2-2*x+3)/(x^2+1):
Figure 1.6.5(b) Graph of gx and y=k π/2,k=1,2,3
Write the equation gx=3 π/2
Context Panel: Solve≻Numerically Solve
The solution of the equation gx=3 π/2 is x^≐0.6400732558, so the location of the vertical asymptote has been determined. The details shown in Figure 1.7.5 have now been confirmed, as has been the conclusion that F is continuous on −∞,x^⋃x^,∞.
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