Question

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)).
2. if (f(x)=\frac{10}{x^{2}}) use a table and your calculator to find (lim_{x

ightarrowinfty}f(x)).

1. given the function (f(x)=2^{x}), find (lim_{x

ightarrowinfty}f(x)).

1. given the function (f(x)=(\frac{1}{2})^{x}), use a table to compute (lim_{x

ightarrowinfty}f(x)).

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. \(\lim_{x\to\infty}f(x)=0\)