To evaluate the limit [tex]\(\lim_{x \rightarrow +\infty}(4x - 7x^2 - 9x^3)\)[/tex], let's analyze the behavior of the polynomial function as [tex]\( x \)[/tex] approaches infinity.
The given function is:
[tex]\[ 4x - 7x^2 - 9x^3 \][/tex]
To understand how this function behaves for very large values of [tex]\( x \)[/tex], it is important to determine which term in the polynomial dominates as [tex]\( x \)[/tex] goes to infinity. In this case, we have:
1. [tex]\( 4x \)[/tex] is a linear term.
2. [tex]\( -7x^2 \)[/tex] is a quadratic term.
3. [tex]\( -9x^3 \)[/tex] is a cubic term.
As [tex]\( x \)[/tex] grows larger and larger, the higher degree terms will have a more significant impact on the overall value of the expression. Specifically, the term with the highest power of [tex]\( x \)[/tex] (the cubic term [tex]\( -9x^3 \)[/tex]) will dominate the behavior of the entire function.
Let's consider the effect of each term individually as [tex]\( x \)[/tex] approaches infinity:
- The linear term [tex]\( 4x \)[/tex] grows linearly (increases as [tex]\( x \)[/tex] increases).
- The quadratic term [tex]\( -7x^2 \)[/tex] grows quadratically but negatively (becomes very large in magnitude negatively as [tex]\( x \)[/tex] increases).
- The cubic term [tex]\( -9x^3 \)[/tex] grows cubically but negatively (becomes very large in magnitude negatively much faster than the other terms as [tex]\( x \)[/tex] increases).
Among these, the cubic term [tex]\( -9x^3 \)[/tex] will grow the fastest in the negative direction. Therefore, the sum of these terms will be dominated by the [tex]\( -9x^3 \)[/tex] term as [tex]\( x \)[/tex] approaches infinity.
Given that [tex]\( -9x^3 \)[/tex] grows without bound negatively as [tex]\( x \rightarrow +\infty \)[/tex], the overall expression [tex]\( 4x - 7x^2 - 9x^3 \)[/tex] will also tend toward negative infinity.
Thus, we conclude:
[tex]\[ \lim_{x \rightarrow +\infty}(4x - 7x^2 - 9x^3) = -\infty \][/tex]
