To evaluate the limit of the given function as [tex]\( x \)[/tex] approaches 2, let's go through the following steps:
[tex]\[ \lim_{{x \to 2}} \left( \frac{5}{3x^4 - 2x^2 - 37} \right)^2 \][/tex]
First, let’s understand the expression inside the limit:
1. The function inside the limit is:
[tex]\[
\left( \frac{5}{3x^4 - 2x^2 - 37} \right)^2
\][/tex]
2. Substitute [tex]\( x = 2 \)[/tex] into the expression [tex]\( 3x^4 - 2x^2 - 37 \)[/tex]:
[tex]\[
3(2)^4 - 2(2)^2 - 37 = 3(16) - 2(4) - 37
\][/tex]
[tex]\[
= 48 - 8 - 37
\][/tex]
[tex]\[
= 48 - 45
\][/tex]
[tex]\[
= 3
\][/tex]
Thus, the expression [tex]\( 3x^4 - 2x^2 - 37 \)[/tex] evaluates to 3 when [tex]\( x = 2 \)[/tex].
3. Now substituting [tex]\( x = 2 \)[/tex] back into the original function:
[tex]\[
\left( \frac{5}{3(2)^4 - 2(2)^2 - 37} \right)^2 = \left( \frac{5}{3} \right)^2
\][/tex]
[tex]\[
= \left( \frac{5}{3} \right)^2
\][/tex]
[tex]\[
= \frac{25}{9}
\][/tex]
Therefore, the evaluated limit is:
[tex]\[ \lim_{{x \to 2}} \left( \frac{5}{3x^4 - 2x^2 - 37} \right)^2 = \frac{25}{9} \][/tex]
