Question

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

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:

\(-\infty\)