find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise…

find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise exist, enter dne.)\n\\( \\lim _ { x \\rightarrow - \\infty } \\left( x ^ { 2 } + 2 x ^ { 9 } \\right) \\)

find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise exist, enter dne.)\n\\( \\lim _ { x \\rightarrow - \\infty } \\left( x ^ { 2 } + 2 x ^ { 9 } \\right) \\)

Answer

Explanation:

Step1: Analyze the leading term

For a polynomial (f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0), as (x\to\pm\infty), the limit is determined by the leading - term (a_nx^n). In the function (y=x^{2}+2x^{9}), the leading term is (2x^{9}) (since the degree of (x^{9}) is higher than the degree of (x^{2}), and for large (|x|), (|2x^{9}|\gg|x^{2}|)).

Step2: Evaluate the limit of the leading term

We know that for (y = 2x^{9}), when (x\to-\infty). Let (t=-x), then (x=-t) and as (x\to-\infty), (t\to+\infty). Substitute (x = - t) into (2x^{9}), we get (2(-t)^{9}=-2t^{9}). Since (\lim_{t\to+\infty}t^{9}=+\infty), then (\lim_{x\to-\infty}2x^{9}=\lim_{t\to+\infty}- 2t^{9}=-\infty).

Answer:

(-\infty)