Chapter 5.5
Notes
Linguistic Mapping Exercises
Knowledge Checks
- Take \(f\) to be the function given by \[f(x) =\begin{cases}\dfrac{x^2-3x-10}{x-5} &\text{if }x\ne 5\\12&\text{if }x = 5.\end{cases}\] Determine \(\lim\limits_{x\to 5}f(x)\) and whether the limit depends on \(f(5).\)
- Take \(f\) to be a function with the property that \(\lim\limits_{x\to 2}f(x)=7\). Determine \(\lim\limits_{x\to 2^-}f(x)\) and \(\lim\limits_{x\to 2^+}f(x).\)
- Show that the following limit does not exist: \[\displaystyle \lim_{x\to 6} \tfrac{|x-6|}{x-6}.\]
- Create a function for which \(\lim\limits_{x\to 3^+}f(x)\) and \(\lim\limits_{x\to 3^-}f(x)\) exist, but \(\lim\limits_{x\to 3}f(x)\) does not exist.
- Determine the way in which the following limits diverge:
- \(\lim\limits_{x\to 6^+} \tfrac{1}{(x-6)^3}\)
- \(\lim\limits_{x\to 6^-} \tfrac{1}{(x-6)^3}\)
- Take \(a\), \(b\), \(c\), and \(d\) to be functions that are defined in an open interval that contains 2 and \[\lim_{x\to 2} a(x) = 6,\quad \lim_{x\to 2} b(x) = -1,\quad \lim_{x\to 2} c(x) = 4,\text {and} \quad \lim_{x\to 2} d(x) = 2.\] 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: \[\displaystyle \lim_{x\to 3}\frac{\sin(4(x-3))}{6(x-3)}.\]
- Use the squeeze theorem to determine the following limit: \[\lim_{x\to 5}\; \sqrt{|x-5|}\sin\left(\frac{1}{x-5}\right).\]
- Take \(f\) to be the function given by \[f(x)=\begin{cases}
\dfrac{x^2+x-2}{x-1}&\text{ if }x<1\\
3x-1 &\text{ if }x\geq 1.
\end{cases}\]
- Determine \(\lim\limits_{x\to 1^-}f(x).\)
- Determine \(\lim\limits_{x\to 1^+}f(x).\)
- Determine whether \(\lim\limits_{x\to 1}f(x)\) exists and determine the limit if it exists.
- Take \(a\) to be a real number and \(f\) to be the function given by \[f(x)=\begin{cases}
x^2-1 &\text{ if }x\leq 4\\
a\log_2(x)+5 &\text{ if }x>4.
\end{cases}\]
- Determine \(\lim\limits_{x\to 4^-}f(x).\)
- Determine \(\lim\limits_{x\to 4^+}f(x).\)
- Determine a value for \(a\) so that \(\lim\limits_{x\to 4}f(x)\) exist.
- Calculate the following:
- \(\lim\limits_{x\to 5}\tfrac{x-1}{|x-5|}\)
- \(\lim\limits_{x \to \infty}\left(\sqrt{16x^2+2x}-4x\right)\)
- Determine all horizontal and vertical asymptotes of the function \(f\) that is given by \[f(x) = \begin{cases} 5^{x}+1&\text{ if }x<-2\\ \dfrac{5x+1}{(x+4)(x-4)} &\text{ if }-2\leq x<4\\ \arctan(-x)+\frac{5}{x+2} &\text{ if }x\geq 4.\end{cases}\]
- 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 \(L\).
Remember, do the knowledge checks before checking the solutions.