To find the limit of the given function as \( x \) approaches infinity, we need to analyze the degrees of the polynomials in the numerator and the denominator. The highest degree term in the numerator is \( 5x^4 \) and in the denominator is \( 12x^2 \). As \( x \) approaches infinity, the lower degree terms become negligible. Therefore, we can simplify the expression by focusing on the highest degree terms.
We need to evaluate the limit
\[
\lim_{x \rightarrow \infty} \frac{5x^4 + 6}{12x^2 + 3}.
\]
As \( x \) approaches infinity, the highest degree terms in the numerator and denominator dominate the behavior of the function. The dominant term in the numerator is \( 5x^4 \) and in the denominator is \( 12x^2 \).
We can simplify the limit by focusing on these dominant terms:
\[
\lim_{x \rightarrow \infty} \frac{5x^4}{12x^2}.
\]
This simplifies to:
\[
\lim_{x \rightarrow \infty} \frac{5}{12} x^{4-2} = \lim_{x \rightarrow \infty} \frac{5}{12} x^2.
\]
As \( x \) approaches infinity, \( \frac{5}{12} x^2 \) approaches infinity. Therefore, the limit is:
\[
\lim_{x \rightarrow \infty} \frac{5x^4 + 6}{12x^2 + 3} = \infty.
\]
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