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Find the limit, or show that it does not exist.


\[\lim_(x\to \infty) \] \left((1-x^2)/(x^3-x+1)\right)

User Frevd
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1 Answer

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Find the following limit...


\lim_(x \to \infty) ((1-x^2)/(x^3-x+1) )

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


\Longrightarrow \lim_(x \to \infty) ((1-x^2)/(x^3-x+1) )

Step 1: Divide everything by the highest power in the denominator, x^3.


\Longrightarrow \lim_(x \to \infty) (((1)/(x^3) -(x^2)/(x^3) )/((x^3)/(x^3) -(x)/(x^3) +(1)/(x^3) ) )

After simplifying we get,


\Longrightarrow \lim_(x \to \infty) (((1)/(x^3) -(1)/(x) )/(1-(1)/(x^2) +(1)/(x^3) ) )

Step 2: Apply
\lim_(x \to a) [(f(x))/(g(x)) ]=( \lim_(x \to a) f(x) )/( \lim_(x \to a) g(x) )


\Longrightarrow( \lim_(x \to \infty) ((1)/(x^3) -(1)/(x) ) )/( \lim_(x \to \infty) (1-(1)/(x^2) +(1)/(x^3)) )

Step 3: Plug in "∞" and solve.


\Longrightarrow( \lim_(x \to \infty) ((1)/((\infty)^3) -(1)/(\infty) ) )/( \lim_(x \to \infty) (1-(1)/((\infty)^2) +(1)/((\infty)^3)) )


\Longrightarrow( \lim_(x \to \infty) (0-0) )/( \lim_(x \to \infty) (1-0+0) )


\Longrightarrow \lim_(x \to \infty) ((0)/(1) ) = \boxed{0}


\Longrightarrow \boxed{\boxed{\lim_(x \to \infty) ((1-x^2)/(x^3-x+1) )=0}} \therefore Sol.

Thus, the limit is solved.

User Giovanni Filardo
by
8.2k points

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