The Limit of f(x)/g(x) When g(c) ≠ 0 for Polynomial Functions: Understanding and Applying Limits in Mathematics

If the functions f and g are polynomials then lim f(x)/g(x) = f(. c)/g(. c) if g(. c) does not equal 0

To understand why the limit of f(x)/g(x) is equal to f(c)/g(c) if g(c) does not equal 0 when f and g are polynomials, we need to understand the concept of limits and apply it to polynomials

To understand why the limit of f(x)/g(x) is equal to f(c)/g(c) if g(c) does not equal 0 when f and g are polynomials, we need to understand the concept of limits and apply it to polynomials.

Let’s start by reviewing the definition of a limit. The limit of a function f(x) as x approaches a value c, denoted as lim(f(x), x→c), gives the value that f(x) approaches as x gets arbitrarily close to c.

In our case, we are considering the function f(x)/g(x) and trying to find the limit of this expression as x approaches a value c. Since both f(x) and g(x) are polynomials, they are continuous functions. This means that as x approaches c, both f(x) and g(x) approach f(c) and g(c), respectively.

Now, if g(c) is not equal to 0, we can divide both the numerator and denominator of f(x)/g(x) by g(x). We obtain:

f(x)/g(x) = [f(x)/g(x)] * [g(x)/g(x)] = [f(x)g(x)]/[g(x)g(x)]

Notice that we have multiplied f(x)/g(x) by 1 in the form of g(x)/g(x), which is valid since g(x) is not equal to 0. Now, if we take the limit of this expression as x approaches c, we obtain:

lim(f(x)/g(x), x→c) = lim([f(x)g(x)]/[g(x)g(x)], x→c)

Since f(x) approaches f(c) and g(x) approaches g(c) as x approaches c, we can substitute those values in the limit:

lim(f(x)/g(x), x→c) = lim([f(x)g(x)]/[g(x)g(x)], x→c) = [f(c)g(c)]/[g(c)g(c)]

Now, since g(c) is not equal to 0, we can divide both the numerator and denominator by g(c):

[f(c)g(c)]/[g(c)g(c)] = [f(c)/g(c)] * [g(c)/g(c)] = [f(c)/g(c)] * 1 = f(c)/g(c)

Therefore, we have shown that if g(c) is not equal to 0 and both f and g are polynomials, then the limit of f(x)/g(x) as x approaches c is equal to f(c)/g(c).

It’s important to note that for this result to hold, g(c) must not be equal to 0, as division by zero is undefined in mathematics.

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