1. given the function, (h(x)=\frac{x - 2}{x+3}), as the values of (x) increase towards infinity, use a table…

1. given the function, (h(x)=\frac{x - 2}{x+3}), as the values of (x) increase towards infinity, use a table to find out what happens to the values of (h(x)).\n4. if (f(x)=\frac{10}{x^{2}}) use a table and your calculator to find (lim_{x\rightarrowinfty}f(x)).\n5. given the function (f(x)=2^{x}), find (lim_{x\rightarrowinfty}f(x)).\n6. given the function (f(x)=(\frac{1}{2})^{x}), use a table to compute (lim_{x\rightarrowinfty}f(x)).

1. given the function, (h(x)=\frac{x - 2}{x+3}), as the values of (x) increase towards infinity, use a table to find out what happens to the values of (h(x)).\n4. if (f(x)=\frac{10}{x^{2}}) use a table and your calculator to find (lim_{x\rightarrowinfty}f(x)).\n5. given the function (f(x)=2^{x}), find (lim_{x\rightarrowinfty}f(x)).\n6. given the function (f(x)=(\frac{1}{2})^{x}), use a table to compute (lim_{x\rightarrowinfty}f(x)).

Answer

Explanation:

Step1: Analyze (h(x)=\frac{x}{x + 3}) as (x\to\infty)

Divide numerator and denominator by (x): (\lim_{x\to\infty}\frac{x}{x + 3}=\lim_{x\to\infty}\frac{1}{1+\frac{3}{x}}). As (x\to\infty), (\frac{3}{x}\to0), so (\lim_{x\to\infty}\frac{1}{1+\frac{3}{x}} = 1).

Step2: Analyze (f(x)=\frac{10}{x^2}) as (x\to\infty)

As (x) gets larger and larger, (x^2) gets very large. So (\frac{10}{x^2}\to0). That is (\lim_{x\to\infty}\frac{10}{x^2}=0).

Step3: Analyze (f(x)=2^x) as (x\to\infty)

The exponential - function (y = a^x) with (a>1) (here (a = 2)) grows without bound as (x\to\infty). So (\lim_{x\to\infty}2^x=\infty).

Step4: Analyze (f(x)=(\frac{1}{2})^x) as (x\to\infty)

The exponential - function (y = a^x) with (0 < a<1) (here (a=\frac{1}{2})) approaches 0 as (x\to\infty). So (\lim_{x\to\infty}(\frac{1}{2})^x = 0).

Answer:

  1. (\lim_{x\to\infty}h(x)=1)
  2. (\lim_{x\to\infty}f(x)=0)
  3. (\lim_{x\to\infty}f(x)=\infty)
  4. (\lim_{x\to\infty}f(x)=0)