Linguistic Mapping Exercises
Knowledge Checks
- Take \(f\) to be the function \[f(x) =\begin{cases}\frac{x^2-3x-10}{x-5} &\text{if }x\ne 5\\12&\text{if }x = 5.\end{cases}\] Determine the following limit and whether the limit depends on \(f(5):\) \[\lim_{x\to 5}f(x).\]
- Take \(f\) to be a function with the property that \(\displaystyle\lim_{x\to 2}f(x)=7\). Determine \(\displaystyle\lim_{x\to 2^-}f(x)\) and \(\displaystyle\lim_{x\to 2^+}f(x).\)
- Show that the following limit does not exist: \[\displaystyle \lim_{x\to 6} \tfrac{|x-6|}{x-6}.\]
- Create a function for which \(\displaystyle\lim_{x\to 3^+}f(x)\) and \(\displaystyle\lim_{x\to 3^-}f(x)\) exist, but \(\displaystyle\lim_{x\to 3}f(x)\) does not exist.
- Determine the way in which the following limits diverge:
- \(\displaystyle \lim_{x\to 6^+} \tfrac{1}{(x-6)^3}\)
- \(\displaystyle \lim_{x\to 6^-} \tfrac{1}{(x-6)^3}\)
- Take \(a\), \(b\), \(c\), and \(d\) to be functions that are defined in an open interval that contains 2 and \[\lim_{x\to 2} a(x) = 6,\quad \lim_{x\to 2} b(x) = -1,\quad \lim_{x\to 2} c(x) = 4,\quad {\rm and} \quad \lim_{x\to 2} d(x) = 2.\] Use the limit laws to determine the following limits: \[\displaystyle \lim_{x\to 2} \left((a(x))^2b(x)+\frac{(x-3)c(x)}{d(x)+3}\right).\]
- Determine the following limit and carefully justify your reasoning: \[\displaystyle \lim_{x\to 3}\tfrac{\sin(4(x-3))}{6(x-3)}.\]
- Use the squeeze theorem to determine the following limit: \[\lim_{x\to 5}\; \sqrt{|x-5|}\sin\big(\tfrac{1}{x-5}\big).\]
- Take \[f(x)=\begin{cases}
\frac{x^2+x-2}{x-1}&\text{ if }x<1\\
3x-1 &\text{ if }x\geq 1.
\end{cases}\]
- Determine \(\displaystyle\lim_{x\to 1^-}f(x)\)
- Determine \(\displaystyle\lim_{x\to 1^+}f(x)\)
- Determine whether \(\displaystyle\lim_{x\to 1}f(x)\) exists. Determine the limit if it exists.
- Take \(a\) to be a real number and \(f\) to be the function \[f(x)=\begin{cases}
x^2-1 &\text{ if }x\leq 4\\
a\log_2(x)+5 &\text{ if }x>4.
\end{cases}\]
- Determine \(\displaystyle\lim_{x\to 4^-}f(x)\)
- Determine \(\displaystyle\lim_{x\to 4^+}f(x)\)
- Find a value of \(a\) for which \(\displaystyle\lim_{x\to 4}f(x)\) exist.
- Calculate the following:
- \(\displaystyle \lim_{x\to 5}\tfrac{x-1}{|x-5|}.\)
- \(\displaystyle \lim_{x \to \infty}\left(\sqrt{16x^2+2x}-4x\right)\)
- Determine all horizontal and vertical asymptotes of the function \(f\) that is given by \[f(x) = \begin{cases} 5^{x}+1&\text{ if }x<-2\\
\frac{5x+1}{(x+4)(x-4)} &\text{ if }-2\leq x<4\\
\arctan(-x)+\frac{5}{x+2} &\text{ if }x\geq 4.\end{cases}\]
- Take \(f\) to be the rational function that is given by \[f(x) = \frac{x^4(x+4)^6(x-1)^2(x-5)^2}{(1-4x)^3(x+3)^{8}}.\] Find a monomial \(g\) so that \(f\) and \(g\) have the same asymptotic behavior at both \(\infty\) and \(-\infty.\)
- Identify a path that describes the position in time of a particle that moves along the line segment \(L\) with endpoints \((2,4)\) and \((5,1)\), has domain equal to \(\mathbb{R}\), is at the midpoint of \(L\) at time \(0\), moves only to the left, and reaches all points of \(L\) except the endpoints of \(L\).