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The expression defining each function consists of a sum or difference of terms and, as written, the long-term behavior of each function is an indeterminate form (an expression which does not have a defined value without additional information) of the type “∞ - ∞”. Find the long-term behavior of each function by first writing each as a ratio of two polynomials.

The Expression Defining Each Function Consists Of A Sum Or Difference Of Terms And As Written The Longterm Behavior Of Each Function Is An Indeterminate Form An class=

Sagot :

Given function is

[tex]y(t)=\frac{t^2+t}{t+2}-\frac{t^3}{t+3}[/tex]

The lease common multiple of (t+2) and (t+3) is (t+2)(t+3) , making the denominator (t+2)(t+3).

[tex]y(t)=\frac{(t^2+t)(t+3)}{(t+2)(t+3)_{}}-\frac{t^3(t+2)}{(t+3)(t+2)}[/tex]

[tex]y(t)=\frac{t^2(t+3)+t(t+3)}{(t+2)(t+3)_{}}-\frac{t^3(t+2)}{(t+3)(t+2)}[/tex]

[tex]y(t)=\frac{t^3+3t^2+t^2+3t}{(t+2)(t+3)_{}}-\frac{t^4+2t^3}{(t+3)(t+2)}[/tex]

[tex]y(t)=\frac{t^3+4t^2+3t-t^4-2t^3}{t\mleft(t+3\mright)+2\mleft(t+3\mright)_{}}[/tex]

[tex]y(t)=\frac{-t^4-t^3+4t^2+3t}{t^2+3t+2t+6_{}}[/tex]

[tex]y(t)=\frac{-t^4-t^3+4t^2+3t}{t^2+5t+6_{}}[/tex]

Hence the ratio of two polynomial is

[tex]y(t)=\frac{-t^4-t^3+4t^2+3t}{t^2+5t+6_{}}[/tex]

The long term behavior is

[tex]y(t)=\frac{-t^4-t^3+4t^2+3t}{t^2+5t+6_{}}\approx\frac{-t^4}{t^2}=-t^2[/tex][tex]\text{ So x }\rightarrow\pm\infty\text{ y likes like y}=-x^2[/tex][tex]\lim _{n\to\infty}-t^2=-\infty[/tex]

[tex]\lim _{n\to-\infty}-t^2=-\infty[/tex]

The is not a horizontal line, So this is not horizontal asymptotes.

Hence the asymptotes is oblique.

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