Question

1. complete the tables and answer the related questions.

as ( x ) gets very large, what do the function values approach?

as ( x ) gets very small, what do the function values approach?

1. consider your responses to question 12. does the function ( h(x) ) have a horizontal asymptote? if so, where?
2. for very large values of ( x ), which term in the numerator and which term in the denominator have the greatest impact on the value of the function ( h(x) )?
3. consider your answer to question 14. how does this information lead to an alternate method for determining the horizontal asymptote?

Explanation:

Step1: Analyze the function for large \(x\)

For \(h(x)=\frac{5x - 10}{x^{2}+x - 6}\), when \(x\) is very large (\(x\to+\infty\)), we can use the fact that for a rational function \(y = \frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\), if \(n

Step2: Analyze the function for very small \(x\) (i.e., \(x\to-\infty\))

When \(x\to-\infty\), for the rational function \(h(x)=\frac{5x - 10}{x^{2}+x - 6}\), since \(n = 1\) and \(m = 2\), \(\lim_{x\to-\infty}\frac{5x - 10}{x^{2}+x - 6}=0\). We calculate \(h(-400)=\frac{5\times(-400)-10}{(-400)^{2}+(-400)-6}\), \(h(-600)=\frac{5\times(-600)-10}{(-600)^{2}+(-600)-6}\), \(h(-800)=\frac{5\times(-800)-10}{(-800)^{2}+(-800)-6}\), \(h(-1000)=\frac{5\times(-1000)-10}{(-1000)^{2}+(-1000)-6}\) by substituting \(x\) values into the function.

Answer:

• For \(x = 400\): \(h(400)=\frac{5\times400 - 10}{400^{2}+400 - 6}=\frac{2000- 10}{160000 + 400-6}=\frac{1990}{160394}\approx0.0124\)
• For \(x = 600\): \(h(600)=\frac{5\times600 - 10}{600^{2}+600 - 6}=\frac{3000-10}{360000+600 - 6}=\frac{2990}{360594}\approx0.0083\)
• For \(x = 800\): \(h(800)=\frac{5\times800 - 10}{800^{2}+800 - 6}=\frac{4000-10}{640000+800 - 6}=\frac{3990}{640794}\approx0.0062\)
• For \(x = 1000\): \(h(1000)=\frac{5\times1000 - 10}{1000^{2}+1000 - 6}=\frac{5000-10}{1000000+1000 - 6}=\frac{4990}{1000994}\approx0.0050\)
• As \(x\) gets very large, the function values approach \(0\)

• For \(x=-400\): \(h(-400)=\frac{5\times(-400)-10}{(-400)^{2}+(-400)-6}=\frac{-2000 - 10}{160000-400 - 6}=\frac{-2010}{159594}\approx - 0.0126\)
• For \(x=-600\): \(h(-600)=\frac{5\times(-600)-10}{(-600)^{2}+(-600)-6}=\frac{-3000-10}{360000-600 - 6}=\frac{-3010}{359394}\approx - 0.0084\)
• For \(x=-800\): \(h(-800)=\frac{5\times(-800)-10}{(-800)^{2}+(-800)-6}=\frac{-4000-10}{640000-800 - 6}=\frac{-4010}{639194}\approx - 0.0063\)
• For \(x=-1000\): \(h(-1000)=\frac{5\times(-1000)-10}{(-1000)^{2}+(-1000)-6}=\frac{-5000-10}{1000000-1000 - 6}=\frac{-5010}{998994}\approx - 0.0050\)
• As \(x\) gets very small, the function values approach \(0\)