question 3 · 1 point\nwhat is the horizontal asymptote of the graph of ( f(x)=\frac{x^{2}+3}{2 x^{2}-7 x}…

question 3 · 1 point\nwhat is the horizontal asymptote of the graph of ( f(x)=\frac{x^{2}+3}{2 x^{2}-7 x} )?\ngive your answer in the form ( y = a ).\nprovide your answer below:

question 3 · 1 point\nwhat is the horizontal asymptote of the graph of ( f(x)=\frac{x^{2}+3}{2 x^{2}-7 x} )?\ngive your answer in the form ( y = a ).\nprovide your answer below:

Answer

Explanation:

Step1: Divide numerator and denominator by (x^{2})

[ \begin{align*} \lim_{x\rightarrow\pm\infty}f(x)&=\lim_{x\rightarrow\pm\infty}\frac{x^{2}+3}{2x^{2}-7x}\ &=\lim_{x\rightarrow\pm\infty}\frac{\frac{x^{2}}{x^{2}}+\frac{3}{x^{2}}}{\frac{2x^{2}}{x^{2}}-\frac{7x}{x^{2}}}\ &=\lim_{x\rightarrow\pm\infty}\frac{1 + \frac{3}{x^{2}}}{2-\frac{7}{x}} \end{align*} ]

Step2: Evaluate the limit

As (x\rightarrow\pm\infty), (\frac{3}{x^{2}}\rightarrow0) and (\frac{7}{x}\rightarrow0) [ \lim_{x\rightarrow\pm\infty}\frac{1 + \frac{3}{x^{2}}}{2-\frac{7}{x}}=\frac{1 + 0}{2-0}=\frac{1}{2} ]

Answer:

(y=\frac{1}{2})