Limits of rational functions can be found by substitution if…
the denominator is not equal to zero
Limits of rational functions can be found by substitution if the function is defined and continuous at the point you are substituting. In other words, if the denominator is not equal to zero at the point where you are substituting, and if the function is not undefined or discontinuous at that point, then you can use substitution to find the limit.
For example, let’s consider the rational function f(x) = (3x^2 – 5x + 2)/(x – 1). If we want to find the limit of this function as x approaches 1, we can simply substitute x = 1 into the function to get:
f(1) = (3(1)^2 – 5(1) + 2)/(1 – 1) = 0/0
Uh oh, we got an indeterminate form! This means that substitution alone won’t work to find the limit at x = 1. We need to use an algebraic method, such as factoring or rationalizing the numerator, to simplify the function and eliminate the zero in the denominator. Once we have done that, we can then use substitution to find the limit.
In summary, substitution can be used to find limits of rational functions if the function is defined and continuous at the point of substitution. Otherwise, we need to use algebraic methods to simplify the function before substitution.
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