Question: User: ②
Quiz # ____
Calculus
Evaluate each of the following
1.) lim
x→0 (x+4)^2 - 16
------------
3x
2.) lim
x→3 √3x
-----
x√x+1
3.) lim
x→2 x - 2
-------
√(x+2) - 2
4.) lim
x→0 3x + 2
-------
x^2 - 2x + 4
5.) lim
x→4 x^3 - 64
---------
x^2 - 16
6.) lim
x→3 x^3 - 13x + 12
---------------
x^3 - 14x + 15
Sir Casy
top-left: a handwritten calculus quiz sheet page with six limit problems arranged in two columns on a blank paper background.
1. **Problem:** Evaluate $$\lim_{x \to 0} \frac{(x+4)^2 - 16}{3x}$$
2. **Formula:** For limits resulting in $$\frac{0}{0}$$, use algebraic simplification or factoring.
3. **Work:**
$$ (x+4)^2 - 16 = (x+4-4)(x+4+4) = x(x+8) $$
So,
$$ \frac{(x+4)^2 - 16}{3x} = \frac{x(x+8)}{3x} $$
Cancel $$x$$:
$$ \frac{\cancel{x}(x+8)}{3\cancel{x}} = \frac{x+8}{3} $$
4. **Evaluate limit:**
$$ \lim_{x \to 0} \frac{x+8}{3} = \frac{0+8}{3} = \frac{8}{3} $$
**Final answer:** $$\frac{8}{3}$$