To evaluate the limit [tex]\(\lim_{x \to 0} \frac{5x^n - 3x^{n-1} + 4x^{n-2}}{2x^n - 6x^{n-2}}\)[/tex], follow these steps:
1. Identify the Dominant Terms:
As [tex]\( x \)[/tex] approaches 0, terms involving lower powers of [tex]\( x \)[/tex] become more significant. Specifically, we need to focus on the smallest power of [tex]\( x \)[/tex] that's present in both the numerator and the denominator.
2. Rewrite with Dominant Terms:
In the numerator [tex]\( 5x^n - 3x^{n-1} + 4x^{n-2} \)[/tex], the dominant term as [tex]\( x \to 0 \)[/tex] is [tex]\( 4x^{n-2} \)[/tex]. In the denominator [tex]\( 2x^n - 6x^{n-2} \)[/tex], the dominant term as [tex]\( x \to 0 \)[/tex] is [tex]\( -6x^{n-2} \)[/tex].
3. Simultaneously Simplify:
Factor [tex]\( x^{n-2} \)[/tex] out of both the numerator and the denominator:
[tex]\[
\frac{5x^n - 3x^{n-1} + 4x^{n-2}}{2x^n - 6x^{n-2}} = \frac{x^{n-2}(5x^2 - 3x + 4)}{x^{n-2}(2x^2 - 6)}
\][/tex]
4. Cancel Out Common Factors:
Since [tex]\( x^{n-2} \)[/tex] is a common factor in both the numerator and the denominator, it can be canceled out (for nonzero [tex]\( x \)[/tex]):
[tex]\[
= \frac{5x^2 - 3x + 4}{2x^2 - 6}
\][/tex]
5. Evaluate the Limit for [tex]\( x \to 0 \)[/tex]:
For [tex]\( x \to 0 \)[/tex], substitute [tex]\( x = 0 \)[/tex] in the simplified expression:
[tex]\[
\lim_{x \to 0} \frac{5x^2 - 3x + 4}{2x^2 - 6} = \frac{5(0)^2 - 3(0) + 4}{2(0)^2 - 6} = \frac{4}{-6} = -\frac{2}{3}
\][/tex]
Thus, the limit is:
[tex]\[
\boxed{-\frac{2}{3}}
\][/tex]
