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Fine the end behavior and x-value of holeEquation is to the right

Fine the end behavior and x-value of holeEquation is to the right-example-1
User Abid Nawaz
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Given the function:


\: f\mleft(x\mright)=(x+1)/(\left(x-3\right)\left(x^2-1\right))

The end behaviour of the function f(x) describes the behaviour of the function as x approaches +∞ and -∞.

When x approaches +∞,


\lim _(x\rightarrow\infty)f(x)=\text{ }\lim _(x\rightarrow\infty)(x+1)/((x-3)(x^2-1))=0

Thus, as x approaches +∞, the function f(x) approaches zero.

When x approaches -∞,


\lim _(x\rightarrow-\infty)f(x)=\text{ }\lim _(x\rightarrow-\infty)(x+1)/((x-3)(x^2-1))=0

Thus, as x approaches -∞, the function f(x) approaches zero.

x-value of the hole:

For a rational function f(x) given as


f(x)=(p(x))/(q(x))

Provided that p(x) and q(x) have a common factor (x-a), the function f(x) will have a hole at x=a.

Thus, from the function f(x)


f(x)=(x+1)/((x-3)(x^2-1))

by expansion, we have


f(x)=\frac{x+1}{(x-3)(x^{}-1)(x+1)}

The expression (x-1) is a common factor of the numerator and the denominator.

Thus,


\begin{gathered} x-1=0 \\ \Rightarrow x=1 \end{gathered}

Hence, the x-value of the hole is 1.

User Cheroky
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