Chapter 2: Problem 3
Which one of the following intervals is not symmetric about \(x=5 ?\) a. (1,9) b. (4,6) c. (3,8) d. (4.5,5.5)
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
The hyperbolic sine function is defined as \(\sinh x=\frac{e^{x}-e^{-x}}{2}\). a. Determine its end behavior by evaluating \(\lim \sinh x\) and \(\lim _{x \rightarrow-\infty} \sinh x\). b. Evaluate sinh 0. Use symmetry and part (a) to sketch a plausible graph for \(y=\sinh x\).
Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=1-\ln x$$
Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$g(x)=e^{1 / x}$$
Sketch a possible graph of a function \(f\) that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes. $$f(-1)=-2, f(1)=2, f(0)=0, \lim _{x \rightarrow \infty} f(x)=1$$, $$\lim _{x \rightarrow-\infty} f(x)=-1$$
a. Use the identity \(\sin (a+h)=\sin a \cos h+\cos a \sin h\) with the fact that \(\lim _{x \rightarrow 0} \sin x=0\) to prove that \(\lim _{x \rightarrow a} \sin x=\sin a\) thereby establishing that \(\sin x\) is continuous for all \(x\). (Hint: Let \(h=x-a \text { so that } x=a+h \text { and note that } h \rightarrow 0 \text { as } x \rightarrow a .)\) b. Use the identity \(\cos (a+h)=\cos a \cos h-\sin a \sin h\) with the fact that \(\lim _{x \rightarrow 0} \cos x=1\) to prove that \(\lim _{x \rightarrow a} \cos x=\cos a.\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
