4.26. lim _(xarrow 5)((1)/(x-5)-(5)/(x^2)-x-20)

Question

Answer 1

To find the limit of the given expression as

approaches 5, we need to simplify the expression first.The expression is:

First, let's factor the denominator in the second term:

So the expression becomes:

Next, we combine the fractions over a common denominator:

Simplify the numerator:

So the expression now is:

As

approaches 5, the denominator \((x-5)\) approaches 0, which means the expression is in an indeterminate form

. To resolve this, we can use L'Hôpital's Rule, which states that if the limit of \(\frac{f(x)}{g(x)}\) as

approaches

is in the form

or

, then:

Let \( f(x) = x-1 \) and \( g(x) = (x-5)(x+4) \).Now, we find the derivatives:

Now apply L'Hôpital's Rule:

Finally, substitute

:

So, the limit is:

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