3) We have
![f(x) = \sec\left(\dfrac{\pi x}2\right) = \dfrac1{\cos\left(\frac{\pi x}2\right)}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Csec%5Cleft%28%5Cdfrac%7B%5Cpi%20x%7D2%5Cright%29%20%3D%20%5Cdfrac1%7B%5Ccos%5Cleft%28%5Cfrac%7B%5Cpi%20x%7D2%5Cright%29%7D)
which has vertical asymptotes (i.e. infinite discontinuities) whenever the denominator is zero. This happens for
![\cos\left(\dfrac{\pi x}2\right) = 0](https://tex.z-dn.net/?f=%5Ccos%5Cleft%28%5Cdfrac%7B%5Cpi%20x%7D2%5Cright%29%20%3D%200)
![\implies \dfrac{\pi x}2 = \cos^{-1}(0) + 2n\pi \text{ or } \dfrac{\pi x}2 = -\cos^{-1}(0) + 2n\pi](https://tex.z-dn.net/?f=%5Cimplies%20%5Cdfrac%7B%5Cpi%20x%7D2%20%3D%20%5Ccos%5E%7B-1%7D%280%29%20%2B%202n%5Cpi%20%5Ctext%7B%20or%20%7D%20%5Cdfrac%7B%5Cpi%20x%7D2%20%3D%20-%5Ccos%5E%7B-1%7D%280%29%20%2B%202n%5Cpi)
(where
is any integer)
![\implies \dfrac{\pi x}2 = \dfrac\pi2 + 2n\pi \text{ or } \dfrac{\pi x}2 = -\dfrac\pi2 + 2n\pi](https://tex.z-dn.net/?f=%5Cimplies%20%5Cdfrac%7B%5Cpi%20x%7D2%20%3D%20%5Cdfrac%5Cpi2%20%2B%202n%5Cpi%20%5Ctext%7B%20or%20%7D%20%5Cdfrac%7B%5Cpi%20x%7D2%20%3D%20-%5Cdfrac%5Cpi2%20%2B%202n%5Cpi)
![\implies x = 1 + 4n \text{ or } x = -1 + 4n](https://tex.z-dn.net/?f=%5Cimplies%20x%20%3D%201%20%2B%204n%20%5Ctext%7B%20or%20%7D%20x%20%3D%20-1%20%2B%204n)
So the graph of
has vertical asymptotes whenever
and
.
4) Given
![h(t) = \begin{cases} t^3+1 & \text{if } t](https://tex.z-dn.net/?f=h%28t%29%20%3D%20%5Cbegin%7Bcases%7D%20t%5E3%2B1%20%26%20%5Ctext%7Bif%20%7D%20t%3C1%20%5C%5C%20%5Cfrac12%20%28t%2B1%29%20%26%20%5Ctext%7Bif%20%7D%20t%5Cge1%20%5Cend%7Bcases%7D)
we have the one-sided limits
![\displaystyle \lim_{t\to1^-} h(t) = \lim_{t\to1} (t^3+1) = 1^3+1 = 2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bt%5Cto1%5E-%7D%20h%28t%29%20%3D%20%5Clim_%7Bt%5Cto1%7D%20%28t%5E3%2B1%29%20%3D%201%5E3%2B1%20%3D%202)
and
![\displaystyle \lim_{t\to1^+} h(t) = \lim_{h\to1} \frac{t+1}2 = \frac{1+1}2 = 1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bt%5Cto1%5E%2B%7D%20h%28t%29%20%3D%20%5Clim_%7Bh%5Cto1%7D%20%5Cfrac%7Bt%2B1%7D2%20%3D%20%5Cfrac%7B1%2B1%7D2%20%3D%201)
The one-sided limits don't match, so the two-sided limit
does not exist. In other words, the limit does not exist at
because the function approaches different values from the left and right side of
.
Answer:
-6a⁷b⁸
Step-by-step explanation:
(-2a³b²)(3a⁴b⁶)
remove parenthesis:
-2a³b²3a⁴b⁶
reorganize the expression:
-2 × 3a³a⁴b²b⁶
simplify:
-6a⁷b⁸
Answer:
I think it may be six , but I may be wrong
Step-by-step explanation:
13.33333333333333333333333333333333 infinity
It should be D. Because the equation is 2(width+ height + length