Chapter 1: Limits
Section 1.6: Continuity
Show that fx=10 x−x2−21 is continuous at the endpoints of its domain.
Control-drag (or type) fx=…
Context Panel: Assign Function
fx=10 x−x2−21→assign as functionf
The graph of f in Figure 1.6.2(a) suggests its domain is the closed interval 3,7, the set of points for which 10 x−x2−21≥0.
The endpoints: fx=0→solvex=3,x=7
The domain 3≤x≤7 can be inferred from:
fx = 10⁢x−x2−21= factor −x−3⁢x−7
Figure 1.6.2(a) Graph of fx=10 x−x2−21
Alternatively, form an inequality in which the radicand is to be nonnegative.
Context Panel: Solve≻Solve
Test for continuity at the endpoints
f3 = 0
limx→3+fx = 0
f7 = 0
limx→7−fx = 0
The function value at x=3 agrees with the limit from the right at x=3. Hence, f is continuous from the right at x=3.
The function value at x=7 agrees with the limit from the left at x=7. Hence, f is continuous from the left at x=7.
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