To determine the vertical asymptote of the function
[tex]\[ f(x) = \frac{3x^4 - 3}{2x - 5} \][/tex]
we need to identify where the denominator of the function equals zero, since division by zero is undefined. The vertical asymptote occurs at these points.
Step-by-step, let's follow this approach:
1. Find the zero of the denominator:
The denominator of the function is [tex]\( 2x - 5 \)[/tex]. To find the value of [tex]\( x \)[/tex] that makes this expression zero, we set:
[tex]\[ 2x - 5 = 0 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
To solve the equation [tex]\( 2x - 5 = 0 \)[/tex], we perform the following steps:
[tex]\[
2x - 5 = 0
\][/tex]
Add 5 to both sides:
[tex]\[
2x = 5
\][/tex]
Divide both sides by 2:
[tex]\[
x = \frac{5}{2}
\][/tex]
Therefore, the vertical asymptote of the function [tex]\( f(x) = \frac{3x^4 - 3}{2x - 5} \)[/tex] is at
[tex]\[ x = \frac{5}{2} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{x = \frac{5}{2}} \][/tex]
