Answer:
2
Step-by-step explanation:
Multiply 4 times 1/2
If you need to find the derivative the answer is 2^-1/2
The answer to this mathmatical problem is 371
Answer:
56
Step-by-step explanation:
corresponding angle to 56 is also opposite to x which means it is also 56
-2in = the equals sign has a slash through it 0
Answer:
![\displaystyle \lim_{x \to -2} \frac{x^3 + 8}{x^4 - 16} = \frac{-3}{8}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20%5Clim_%7Bx%20%5Cto%20-2%7D%20%5Cfrac%7Bx%5E3%20%2B%208%7D%7Bx%5E4%20-%2016%7D%20%3D%20%5Cfrac%7B-3%7D%7B8%7D)
General Formulas and Concepts:
<u>Calculus</u>
Limits
Limit Rule [Constant]: ![\displaystyle \lim_{x \to c} b = b](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20c%7D%20b%20%3D%20b)
Limit Rule [Variable Direct Substitution]: ![\displaystyle \lim_{x \to c} x = c](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20c%7D%20x%20%3D%20c)
Limit Property [Addition/Subtraction]: ![\displaystyle \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20c%7D%20%5Bf%28x%29%20%5Cpm%20g%28x%29%5D%20%3D%20%20%5Clim_%7Bx%20%5Cto%20c%7D%20f%28x%29%20%5Cpm%20%5Clim_%7Bx%20%5Cto%20c%7D%20g%28x%29)
L'Hopital's Rule
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Addition/Subtraction]: ![\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%20%2B%20g%28x%29%5D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bg%28x%29%5D)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Step-by-step explanation:
We are given the following limit:
![\displaystyle \lim_{x \to -2} \frac{x^3 + 8}{x^4 - 16}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20%5Clim_%7Bx%20%5Cto%20-2%7D%20%5Cfrac%7Bx%5E3%20%2B%208%7D%7Bx%5E4%20-%2016%7D)
Let's substitute in <em>x</em> = -2 using the limit rule:
![\displaystyle \lim_{x \to -2} \frac{x^3 + 8}{x^4 - 16} = \frac{(-2)^3 + 8}{(-2)^4 - 16}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20%5Clim_%7Bx%20%5Cto%20-2%7D%20%5Cfrac%7Bx%5E3%20%2B%208%7D%7Bx%5E4%20-%2016%7D%20%3D%20%5Cfrac%7B%28-2%29%5E3%20%2B%208%7D%7B%28-2%29%5E4%20-%2016%7D)
Evaluating this, we arrive at an indeterminate form:
![\displaystyle \lim_{x \to -2} \frac{x^3 + 8}{x^4 - 16} = \frac{0}{0}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20%5Clim_%7Bx%20%5Cto%20-2%7D%20%5Cfrac%7Bx%5E3%20%2B%208%7D%7Bx%5E4%20-%2016%7D%20%3D%20%5Cfrac%7B0%7D%7B0%7D)
Since we have an indeterminate form, let's use L'Hopital's Rule. Differentiate both the numerator and denominator respectively:
![\displaystyle \lim_{x \to -2} \frac{x^3 + 8}{x^4 - 16} = \lim_{x \to -2} \frac{3x^2}{4x^3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-2%7D%20%5Cfrac%7Bx%5E3%20%2B%208%7D%7Bx%5E4%20-%2016%7D%20%3D%20%5Clim_%7Bx%20%5Cto%20-2%7D%20%5Cfrac%7B3x%5E2%7D%7B4x%5E3%7D)
Substitute in <em>x</em> = -2 using the limit rule:
![\displaystyle \lim_{x \to -2} \frac{3x^2}{4x^3} = \frac{3(-2)^2}{4(-2)^3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-2%7D%20%5Cfrac%7B3x%5E2%7D%7B4x%5E3%7D%20%3D%20%5Cfrac%7B3%28-2%29%5E2%7D%7B4%28-2%29%5E3%7D)
Evaluating this, we get:
![\displaystyle \lim_{x \to -2} \frac{3x^2}{4x^3} = \frac{-3}{8}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-2%7D%20%5Cfrac%7B3x%5E2%7D%7B4x%5E3%7D%20%3D%20%5Cfrac%7B-3%7D%7B8%7D)
And we have our answer.
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Limits