For a rational function f(x)/g(x), I know that we can say it is continuous if both f(x) and g(x) is continuous while g(x) is also not 0.

So my question is: for the same rational function f(x)/g(x), can we say it is continuous if we know the limit of f(x) exists and equals to 0 at our point of interest while we know the denominator g(x) is not zero but it is dis-continuous? My doubt is: the rational function would be continuous and reach 0 anyway because the numerator is 0, so in this scenario would the continuity of the denominator affect the continuity of the rational function if the denominator is not zero.

Hope you understand my question. Thank you.

So my question is: for the same rational function f(x)/g(x), can we say it is continuous if we know the limit of f(x) exists and equals to 0 at our point of interest while we know the denominator g(x) is not zero but it is dis-continuous? My doubt is: the rational function would be continuous and reach 0 anyway because the numerator is 0, so in this scenario would the continuity of the denominator affect the continuity of the rational function if the denominator is not zero.

Hope you understand my question. Thank you.

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