91Ó°ÊÓ

Prove that $$\lim _{x \rightarrow \infty} f(x)=\lim _{t \rightarrow 0^{+}} f(1 / t)$$ and $$\lim _{x \rightarrow-\infty} f(x)=\lim _{t \rightarrow 0^{-}} f(1 / t)$$ if these limits exist.

To prove that sine has the Direct Substitution Property we need to show that \(\lim _{x \rightarrow a} \sin x=\sin a\) for every real number \(a .\) If we let \(h=x-a,\) then \(x=a+h\) and \(x \rightarrow a \Longleftrightarrow h \rightarrow 0 .\) So an equivalent statement is that $$\lim _{h \rightarrow 0} \sin (a+h)=\sin a$$ Use \(\square\) to show that this is true.

Determine whether \(f\) is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. $$f(x)=\frac{x}{x^{2}+1}$$

Prove that cosine has the Direct Substitution Property.

Determine whether \(f\) is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. $$f(x)=\frac{x^{2}}{x^{4}+1}$$

Determine whether \(f\) is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. $$f(x)=\frac{x}{x+1}$$

Let \(f\) and \(g\) be linear functions with equations \(f(x)=m_{1} x+b_{1}\) and \(g(x)=m_{2} x+b_{2} .\) Is \(f \circ g\) also a linear function? If so, what is the slope of its graph?

If $$\lim _{x \rightarrow 1} \frac{f(x)-8}{x-1}=10,$$ find $$\lim _{x \rightarrow 1} f(x)$$

If you invest \(x\) dollars at 4\(\%\) interest compounded annually, then the amount \(A(x)\) of the investment after one year is \(A(x)=1.04 x .\) Find \(A \circ A, A \circ A \circ A,\) and \(A \circ A \circ A \circ A\) What do these compositions represent? Find a formula for the composition of \(n\) copies of \(A .\)

Prove that if \(\lim _{x \rightarrow a} g(x)=0\) and \(\lim _{x \rightarrow a} f(x)\) exists and is not \(0,\) then $$\lim _{x \rightarrow a} \frac{f(x)}{g(x)} \quad$$ does not exist