Question

problem. 4 : determine the limit. \\(\lim\limits_{x \to 7^+} \frac{3}{x - 7} - \ln(x - 7) = \\) \\(\infty\\) \
problem. 5 : consider the rational function \\(f(x) = \frac{x^2 + 2x}{x^2 - 4}\\). identify any vertical asymptotes. \\(x = 2\\) \
problem. 5.1 : are there any holes? if so, give the coordinates of the hole. if not, enter
one\. \\((-2, 1/2)\\) \
problem. 5.1.1 : identify any horizontal asymptotes. if none, enter
one\. \\(y = \\)

Explanation:

Step1: Analyze the degrees of numerator and denominator

For the rational function \( f(x)=\frac{x^{2}+2x}{x^{2}-4} \), the degree of the numerator (highest power of \( x \)) is \( 2 \) (from \( x^{2} \)) and the degree of the denominator is also \( 2 \) (from \( x^{2} \)).

Step2: Use the rule for horizontal asymptotes of rational functions

When the degrees of the numerator (\( n \)) and denominator (\( m \)) are equal (\( n = m \)), the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator \( x^{2}+2x \) is \( 1 \) (coefficient of \( x^{2} \)), and the leading coefficient of the denominator \( x^{2}-4 \) is also \( 1 \) (coefficient of \( x^{2} \)). So the horizontal asymptote is \( y=\frac{1}{1}=1 \).

Answer:

\( 1 \)