What is the continuity of #f(t) = 3 - sqrt(9-t^2)#?

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Answer 1 · 2015-05-01T15:42:03

#f(t) = 3 - sqrt(9-t^2)# has domain #[-3,3]#

For #a# in #(-3,3)#, #lim_(trarra) f(t) = f(a)# because

#lim_(trarra)(3 - sqrt(9-t^2)) = 3-lim_(trarra) sqrt(9-t^2)#

#= 3-sqrt(lim_(trarra) (9-t^2)) = 3-sqrt(9-lim_(trarra) t^2))#

#=3-sqrt(9-a^2) = f(a)#

So #f# is continuous on #(-3,3)#.

Similar reasoning will show that

#lim_(trarr-3^+) f(t) = f(-3)# and

#lim_(trarr3^-) f(t) = f(3)#

So #f# is continuous on #[-3,3]#.