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:
The answer is option 1.
Step-by-step explanation:
Given that the formula for tangent is tanθ = opposite/adjacent :
![\tan(θ) = \frac{oppo.}{adj.}](https://tex.z-dn.net/?f=%20%5Ctan%28%CE%B8%29%20%20%3D%20%20%5Cfrac%7Boppo.%7D%7Badj.%7D%20)
Let oppo. = BC = 16,
Let adj. = AB = 12,
![\tan(θ) = \frac{16}{12}](https://tex.z-dn.net/?f=%20%5Ctan%28%CE%B8%29%20%20%3D%20%20%5Cfrac%7B16%7D%7B12%7D%20)
Answer:
its thrid one
Step-by-step explanation:
4x² + 3x + 5 = 0
x = <u>-3 +/- √(3² - 4(4)(5))</u>
2(4)
x = <u>-3 +/- √(9 - 80)</u>
8
x = <u>-3 +/- √(-71)
</u> 8<u>
</u>x = <u>-3 +/- √(71 × (-1))</u>
8
x = <u>-3 +/- √(71) × √(-1)
</u> 8<u>
</u>x = <u>-3 +/- 8.43i
</u> 8
x = -0.375 +/- 1.05375i
x = -0.375 + 1.05375i x = -0.375 - 1.05375i
<u />
Answer: It would be 170
Step-by-step explanation:
5 x 10 + 5 x 10 + 5 x 10 = 150 but since it is park of the rectangle and not the triangle you don't multiply by 1/2 but the triangle part 4 x 5 you do 4 x 5 x 1/2 with is ten then you do it for the other side 4 x 5 x 1/2 equals ten. So 150 + 10 + 10=170