How do you use the definition of continuity and the properties of limits to show that the function #f(x)=x^2 + sqrt(7-x)# is continuous at a=4?

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Answer 1 · 2017-02-09T21:31:31

see below

#f(x)=x^2+sqrt(7-x), a=4#

We need to show that #f(4)=lim_(x->4) x^2+sqrt(7-x)#

That is,

#f(4)=16+sqrt(7-4) = 16+sqrt3#

#lim_(x->4) x^2+sqrt(7-x) = lim_(x->4) x^2+lim_(x->4) sqrt(7-x) #

#= lim_(x->4) x^2+sqrt(lim_(x->4)(7-x)#

#= lim_(x->4) x^2+sqrt(lim_(x->4)7-lim_(x->4)x)#

#=16+sqrt(7-4)#

#=16+sqrt3#

Since #f(4)=lim_(x->4) x^2+sqrt(7-x)=16+sqrt3# therefore

#f(x)=x^2+sqrt(7-x)# is continuous at a = 4