If #lim_(xrarr2)(f(x)-4)/(x-2)=C# for constant C. Find #lim_(xrarr2)#f(x). What theorems were used to come to conclusion?

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Answer 1 · 2018-04-28T00:53:51

4

We can use the properties

#lim_{x to a}[f(x)+g(x)] = lim_{x to a} f(x) +lim_{x to a}g(x)#

and

#lim_{x to a}[f(x)g(x)] = lim_{x to a} f(x) lim_{x to a}g(x)#

provided both the limits on the right exists.

Thus

#lim_{x to 2}(f(x)-4) = lim_{x to 2} (f(x)-4)/(x-2) lim_{x to 2}(x-2)#
#qquad qquad qquad qquad = C times 0 = 0#

Then

#lim_{x to 2}f(x) = lim_{x to 2}(f(x)-4)+lim_{x to 2}4#
#qquad = 0+4 = 4#