Quotient rule for limits
The quotient rule for limits is a rule used to find the limit of a quotient of two functions as the input approaches a certain value
The quotient rule for limits is a rule used to find the limit of a quotient of two functions as the input approaches a certain value. It is specifically used when applying limits to fractions where the numerator and denominator both approach zero or both approach infinity.
The quotient rule states that if the functions f(x) and g(x) are both continuous at a point c, and g(c) is not equal to zero, then the limit of the quotient of f(x) and g(x) as x approaches c can be found by taking the limit of the numerator divided by the limit of the denominator:
lim(x→c) [f(x) / g(x)] = [lim(x→c) f(x)] / [lim(x→c) g(x)]
In other words, if the individual limits of the numerator and denominator exist, then the limit of their quotient exists as well.
It is important to note that the quotient rule only applies when g(c) is not equal to zero. If g(c) equals zero, then the limit of the quotient does not exist, and we must use other methods to determine the limit.
Using the quotient rule for limits can be particularly useful when dealing with functions that are not easily simplified or evaluated. By breaking down the expression into separate limits of the numerator and denominator, it becomes easier to determine the overall limit.
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