9. find all intercepts and asymptotes of the following function, clearly label each piece of…

Explanation:

Step1: Find the (x -)intercepts

Set (y = f(x)=0), so (\frac{5x^{2}}{x^{4}-625}=0). Since the numerator (5x^{2}=0) when (x = 0) (and (x = 0) does not make the denominator (x^{4}-625=(0)^{4}-625=- 625\neq0)).

Step2: Find the (y -)intercept

Set (x = 0), then (y=f(0)=\frac{5\times(0)^{2}}{(0)^{4}-625}=0)

Step3: Find the vertical asymptotes

Set the denominator (x^{4}-625=(x^{2}+25)(x^{2}-25)=(x^{2}+25)(x + 5)(x - 5)=0). Since (x^{2}+25>0) for all real (x), the vertical asymptotes occur at (x=-5) and (x = 5) (because these values make the denominator zero and the numerator non - zero).

Step4: Find the horizontal asymptote

Since the degree of the numerator (n = 2) and the degree of the denominator (m=4) and (n<m). Using the rule for rational functions (\lim_{x\rightarrow\pm\infty}\frac{5x^{2}}{x^{4}-625}=\lim_{x\rightarrow\pm\infty}\frac{\frac{5x^{2}}{x^{4}}}{\frac{x^{4}}{x^{4}}-\frac{625}{x^{4}}}=\lim_{x\rightarrow\pm\infty}\frac{\frac{5}{x^{2}}}{1-\frac{625}{x^{4}}}=0)

Answer:

• (x -)intercept: (x = 0)
• (y -)intercept: (y = 0)
• Vertical asymptotes: (x=-5) and (x = 5)
• Horizontal asymptote: (y = 0)