Question No.1 - Easy (1 point for each function)
Version 1
Find \(\lim_{x\to0}f(x),\mathrm{~where~}x+1\leq f(x)\leq e^x\) for all \(x\) in the interval (-1,1).
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer. No decimal numbers as a final answer.
Version 2
Find \(\lim_{x\to1}h(x),\mathrm{~where~}1-2x\leq h(x)\leq-x^{2}\) for all \(x\) in the interval (0,2).
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer. No decimal numbers as a final answer.
Version 3
Find \(\lim_{x\to0}h(x),\mathrm{~where~}e^{-x}\leq h(x)\leq-x+1\) for all \(x\) in the interval (-1,1).
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer. No decimal numbers as a final answer.
Version 4
Find \(\lim_{x\to1}f(x),\mathrm{~where~}2x-1\leq f(x)\leq x^{2}\) for all \(x\) in the interval (0,2).
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer. No decimal numbers as a final answer.
Question No.2 - Medium (3 points - 1.5 points for each part)
Version 1
Assume that \(f(x)\) is a continuous function for all real numbers. Determine the following limits if
\[\lim_{x\to0^+}f(x)=A\ \mathrm{and} \lim_{x\to0^-}f(x)=B :\]
Part (A): \(\lim_{x\to0^-}\left(f(x^3)-f(x)\right)\)
Part (B): \(\lim_{x\to0^-}f(x^2-x)\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.
Version 2
Assume that \(g(x)\) is a continuous function for all real numbers. Determine the following limits if
\[\lim_{x\to0^+}g(x)=C\ \mathrm{and} \lim_{x\to0^-}g(x)=D :\]
Part (A): \(\lim_{x\to0^+}\left(g(x^2)-g(x)\right)\)
Part (B): \(\lim_{x\to0^+}g(x^3-x)\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.
Version 3
Assume that \(f(x)\) is a continuous function for all real numbers. Determine the following limits if \(\lim_{x\to0^+}f(x)=C\ \mathrm{and} \lim_{x\to0^-}f(x)=D :\)
Part (A): \(\lim_{x\to0^-}\left(f(x^3)-f(x)\right)\)
Part (B): \(\lim_{x\to0^-}f(x^2-x)\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.
Version 4
Assume that \(g(x)\) is a continuous function for all real numbers. Determine the following limits if \(\lim_{x\to0^+}g(x)=A\ \mathrm{and} \lim_{x\to0^-}g(x)=B{:}\)
Part (A): \(\lim_{x\to0^+}\left(g(x^2)-g(x)\right)\)
Part (B): \(\lim_{x\to0^+}g(x^3-x)\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.
Question No.3 - Medium (3 points - 1 point for each part)
Version 1
Find all horizontal asymptotes, if any, of the function \(f(x)=\sqrt{x}(\sqrt{x+3}-\sqrt{x-2}).\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.
Version 2
Find all horizontal asymptotes, if any, of the function \(f(x)=\sqrt{x}(\sqrt{2x+1}-\sqrt{2x-1}).\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.
Version 3
Find all horizontal asymptotes, if any, of the function \(f(x)=\sqrt{x}(\sqrt{3x+7}-\sqrt{3x-7}).\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.
Version 4
Find all horizontal asymptotes, if any, of the function \(f(x)=\sqrt{x}(\sqrt{x+5}-\sqrt{x-2}).\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.
Question No.4 - Difficult (4 points)
Version 1
For what values of \(a\) and \(b\) is the following function continuous for all real numbers?
\(g(x)=\left\{\begin{array}{ll}b-\frac{a(x+1)}{\mid1-x\mid}&\quad\text{for} x\leq-1\\4\arctan(x)+a&\quad\text{for} -1<x<1\\3be^x-a\ln(x)&\quad\text{for}\ x \geq1\end{array}\right.\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer for \(a\) and \(b\).
Version 2
For what values of \(a\) and \(b\) is the following function continuous for all real numbers?
\(f(x)=\left\{\begin{array}{cl}\frac{a\cos(x)}{3}+b-2&\quad\text{for} x<0\\a+\frac{b\mid x-2\mid}{x-2}&\quad\text{for}\ 0\le x<2\\3x^2-7b&\quad\text{for}\ x\ge2\end{array}\right.\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer for \(a\) and \(b\).
Version 3
For what values of \(m\) and \(n\) is the following function continuous for all real numbers?
\(f(x)=\left\{\begin{array}{ll}-x^2+3&\quad\text{for}\ x<-1\\2n+\frac{m(x+1)}{\mid x+1\mid}&\quad\text{for} -1\le x<0\\-\frac{m\cos(x)}{2}+n+1&\quad\text{for}\ x\ge0\end{array}\right.\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer for \(m\) and \(n\).
Version 4
For what values of \(m\) and \(n\) is the following function continuous for all real numbers?
\(\left.g(x)=\left\{\begin{array}{ll}m+\frac{n(x-3)}{\mid3-x\mid}&\quad\text{for}\ x<0\\-4m\arctan(x)+ne^{2x}&\quad\text{for}\ 0\leq x\leq1\\3n+m\ln(x)+1&\quad\text{for}\ x>1\end{array}\right.\right.\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer for \(m\) and \(n\).
Question No.5 - Challenging (4 points)
Version 1
Show that the equation \(4^x=\frac{34}x\) has a solution for \(\mathrm{x>0}\).
Make sure to verify all assumptions.
You must clearly and coherently justify your work.
Version 2
Show that the equation \(3^x=\frac{20}{x}\) has a solution for \(\mathrm{x>0}\).
Make sure to verify all assumptions.
You must clearly and coherently justify your work.
Version 3
Show that the equation \(1 5^{x}=\frac{60}{x}\) has a solution for \(\mathrm{x>0}\).
Make sure to verify all assumptions.
You must clearly and coherently justify your work.
Version 4
Show that the equation \(2^x=\frac{10}x\) has a solution for \(\mathrm{x>0}\).
Make sure to verify all assumptions.
You must clearly and coherently justify your work.