The end behavior of [tex]\lim_{x \to \infty} \frac{-5x^{4} +13x}{x^{3} }[/tex] is as x tends to ∞, h(x) is -∞.
What is limit of functions?
A function may approach two different limits. One where the variable approaches its limit through values larger than the limit and the other where the variable approaches its limit through values smaller than the limit. In such a case, the limit is not defined but the right and left-hand limit exist. The right-hand limit of a function is the value of the function approaches when the variable approaches its limit from the right.
Given Limit function
[tex]\lim_{x \to \infty} \frac{-5x^{4} +13x}{x^{3} }[/tex]
= [tex]\lim_{x \to \infty} \frac{x^{4} (-5+\frac{13}{x^{3} }) }{x^{3} }[/tex]
= [tex]\lim_{x\to \infty} x(-5+\frac{13}{x^{3} } )[/tex]
= [tex]\lim_{x \to \infty} -5x+\frac{13}{x^{2} }[/tex]
= - ∞ + 0
= - ∞
Thus, the end behavior of [tex]\lim_{x \to \infty} \frac{-5x^{4} +13x}{x^{3} }[/tex] is as x tends to ∞, h(x) is -∞.
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