By applying the exponential and logarithmic functions, we have
![x^{\sin(x)} = \exp \left(\ln \left(x^{\sin(x)}\right)\right)](https://tex.z-dn.net/?f=x%5E%7B%5Csin%28x%29%7D%20%3D%20%5Cexp%20%5Cleft%28%5Cln%20%5Cleft%28x%5E%7B%5Csin%28x%29%7D%5Cright%29%5Cright%29)
Then in the limit,
![\displaystyle \lim_{x\to0} x^{\sin(x)} = \lim_{x\to0} \exp \left(\ln \left(x^{\sin(x)}\right)\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%5Cto0%7D%20x%5E%7B%5Csin%28x%29%7D%20%3D%20%5Clim_%7Bx%5Cto0%7D%20%20%5Cexp%20%5Cleft%28%5Cln%20%5Cleft%28x%5E%7B%5Csin%28x%29%7D%5Cright%29%5Cright%29)
![\displaystyle \lim_{x\to0} x^{\sin(x)} = \exp \left( \lim_{x\to0} \ln \left(x^{\sin(x)}\right)\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%5Cto0%7D%20x%5E%7B%5Csin%28x%29%7D%20%3D%20%20%5Cexp%20%5Cleft%28%20%5Clim_%7Bx%5Cto0%7D%20%5Cln%20%5Cleft%28x%5E%7B%5Csin%28x%29%7D%5Cright%29%5Cright%29)
![\displaystyle \lim_{x\to0} x^{\sin(x)} = \exp \left( \lim_{x\to0} \sin(x) \ln(x) \right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%5Cto0%7D%20x%5E%7B%5Csin%28x%29%7D%20%3D%20%20%5Cexp%20%5Cleft%28%20%5Clim_%7Bx%5Cto0%7D%20%5Csin%28x%29%20%5Cln%28x%29%20%5Cright%29)
![\displaystyle \lim_{x\to0} x^{\sin(x)} = \exp \left( \lim_{x\to0} \frac{\ln(x)}{\csc(x)} \right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%5Cto0%7D%20x%5E%7B%5Csin%28x%29%7D%20%3D%20%20%5Cexp%20%5Cleft%28%20%5Clim_%7Bx%5Cto0%7D%20%5Cfrac%7B%5Cln%28x%29%7D%7B%5Ccsc%28x%29%7D%20%5Cright%29)
As x approaches 0 (from the right), both ln(x) and csc(x) approach infinity (ignoring sign). Applying L'Hopitâl's rule gives
![\displaystyle \lim_{x\to0} x^{\sin(x)} = \exp \left( \lim_{x\to0} \frac{\frac1x}{-\csc(x)\cot(x)} \right) = \exp \left( -\lim_{x\to0} \frac{\sin^2(x)}{x\cos(x)} \right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%5Cto0%7D%20x%5E%7B%5Csin%28x%29%7D%20%3D%20%20%5Cexp%20%5Cleft%28%20%5Clim_%7Bx%5Cto0%7D%20%5Cfrac%7B%5Cfrac1x%7D%7B-%5Ccsc%28x%29%5Ccot%28x%29%7D%20%5Cright%29%20%3D%20%20%5Cexp%20%5Cleft%28%20-%5Clim_%7Bx%5Cto0%7D%20%5Cfrac%7B%5Csin%5E2%28x%29%7D%7Bx%5Ccos%28x%29%7D%20%5Cright%29)
Recall that
![\displaystyle \lim_{x\to0} \frac{\sin(x)}{x} = 1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%5Cto0%7D%20%5Cfrac%7B%5Csin%28x%29%7D%7Bx%7D%20%3D%201)
Then
![\displaystyle \lim_{x\to0} \frac{\sin^2(x)}{x\cos(x)} = \lim_{x\to0} \frac{\sin(x)}{\cos(x)} = \lim_{x\to0} \tan(x) = 0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%5Cto0%7D%20%5Cfrac%7B%5Csin%5E2%28x%29%7D%7Bx%5Ccos%28x%29%7D%20%3D%20%20%5Clim_%7Bx%5Cto0%7D%20%5Cfrac%7B%5Csin%28x%29%7D%7B%5Ccos%28x%29%7D%20%3D%20%5Clim_%7Bx%5Cto0%7D%20%5Ctan%28x%29%20%3D%200)
So, our limit is
![\displaystyle \lim_{x\to0} x^{\sin(x)} = \exp(0) = \boxed{1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%5Cto0%7D%20x%5E%7B%5Csin%28x%29%7D%20%3D%20%20%5Cexp%280%29%20%3D%20%5Cboxed%7B1%7D)
Answer:
I belive the first one is 5:7 And the second one is 7:12
Step-by-step explanation:
Answer: I would say A or C bc they are similar but C seems more reasonable.
Step-by-step explanation:
Answer:
x = 18 degrees
Step-by-step explanation:
We are given 72 degrees. The box on the bottom left of the ramp is at a 90 degree angle. Given that this triangle will add up to 180 degrees no matter what we add the two given degrees; 72 + 90 = 162. Now we subtract; 180 - 162 = 18 degrees.
Answer: 19
Step-by-step explanation:
Count all the green X's. The numbers at the bottom just talk about how many cookie boxes they each sold.