Since the limit becomes the undetermined form
![\displaystyle \lim_{x\to 1} \dfrac{x^3-2x^2+3x-2}{2x^4-3x+1} \to \dfrac{0}{0}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%5Clim_%7Bx%5Cto%201%7D%20%5Cdfrac%7Bx%5E3-2x%5E2%2B3x-2%7D%7B2x%5E4-3x%2B1%7D%20%5Cto%20%5Cdfrac%7B0%7D%7B0%7D%20)
it means that both polynomials have a root at
. So, we can fact both numerator and denominator:
![x^3-2x^2+3x-2 = (x-1)(x^2-x+2)](https://tex.z-dn.net/?f=%20x%5E3-2x%5E2%2B3x-2%20%3D%20%28x-1%29%28x%5E2-x%2B2%29%20)
![2x^4-3x+1 = (x-1)(2x^3+2x^2+2x-1)](https://tex.z-dn.net/?f=%202x%5E4-3x%2B1%20%3D%20%28x-1%29%282x%5E3%2B2x%5E2%2B2x-1%29%20)
So, the fraction becomes
![\dfrac{(x-1)(x^2-x+2)}{(x-1)(2x^3+2x^2+2x-1)} = \dfrac{x^2-x+2}{2x^3+2x^2+2x-1}](https://tex.z-dn.net/?f=%20%5Cdfrac%7B%28x-1%29%28x%5E2-x%2B2%29%7D%7B%28x-1%29%282x%5E3%2B2x%5E2%2B2x-1%29%7D%20%3D%20%5Cdfrac%7Bx%5E2-x%2B2%7D%7B2x%5E3%2B2x%5E2%2B2x-1%7D%20)
Now, as x approaches 1, you have no problems anymore:
![\displaystyle \lim_{x\to 1} \dfrac{x^2-x+2}{2x^3+2x^2+2x-1} \to \dfrac{2}{5}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%5Clim_%7Bx%5Cto%201%7D%20%5Cdfrac%7Bx%5E2-x%2B2%7D%7B2x%5E3%2B2x%5E2%2B2x-1%7D%20%5Cto%20%5Cdfrac%7B2%7D%7B5%7D%20)
Ok so our original fraction is:
![\frac{36 {x}^{12} }{6 {x}^{9} }](https://tex.z-dn.net/?f=%20%5Cfrac%7B36%20%7Bx%7D%5E%7B12%7D%20%7D%7B6%20%7Bx%7D%5E%7B9%7D%20%7D%20)
To simplify this fraction, look for instances where the values on the top and bottom can be reduced:
For example, 36 over 6 is the same as 6 over 1, so we can simplify the fraction so it is:
![\frac{6 {x}^{12} }{ {x}^{9} }](https://tex.z-dn.net/?f=%20%5Cfrac%7B6%20%7Bx%7D%5E%7B12%7D%20%7D%7B%20%7Bx%7D%5E%7B9%7D%20%7D%20)
We can also eliminate the denominator by dividing the nominator by x^9 so:
![\frac{6 {x}^{12} }{ {x}^{9} } \div \frac{ {x}^{9} }{ {x}^{9} }](https://tex.z-dn.net/?f=%20%5Cfrac%7B6%20%7Bx%7D%5E%7B12%7D%20%7D%7B%20%7Bx%7D%5E%7B9%7D%20%7D%20%20%5Cdiv%20%20%5Cfrac%7B%20%7Bx%7D%5E%7B9%7D%20%7D%7B%20%7Bx%7D%5E%7B9%7D%20%7D%20)
![= 6 {x}^{3}](https://tex.z-dn.net/?f=%20%3D%206%20%7Bx%7D%5E%7B3%7D%20)
And that is the simplified answer of the fraction
Hope this helped
Answer:
12 and 13
Step-by-step explanation:
I hope this helps! Have a lovely day!! :)
Answer:
No, the expression is not linear because the highest power of x is 2.
Answer:
It has to be a single digit between 1 and 10