Show by example that although f and g may fail to have a limit at a cluster point c, their sum f+g
can have a limit at c. Give similar examples for the product fg and for the quotient f/g. Please
provide detailed answers.
Solution
let f(x)= sec^2x let g(x) = -tan^2x then f+g will give constant \'1\' Notice that these
do not limit at pi/2 but f+g does have a limit everywhere. Similarly for product, f=tan(x) does
not exist around pi/2, but when multiplied with g =cos(x) , they give value sin(x), which has
limit everywhere Similarly, g(x)=(1/x) has no limit at x=0, f(x) = x, but f/g = x^2 has limit
everywhere..
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Show by example that although f and g may fail to have a limit at a .pdf
1. Show by example that although f and g may fail to have a limit at a cluster point c, their sum f+g
can have a limit at c. Give similar examples for the product fg and for the quotient f/g. Please
provide detailed answers.
Solution
let f(x)= sec^2x let g(x) = -tan^2x then f+g will give constant '1' Notice that these
do not limit at pi/2 but f+g does have a limit everywhere. Similarly for product, f=tan(x) does
not exist around pi/2, but when multiplied with g =cos(x) , they give value sin(x), which has
limit everywhere Similarly, g(x)=(1/x) has no limit at x=0, f(x) = x, but f/g = x^2 has limit
everywhere.
