+42 Show that \(f(x) = x\sqrt{x+3}\) is continuous at the point x = -1 using the definition of continuity. (It must show the three conditions of continuity satisfied)
So far I've boiled it down to:
f(-1) = ??
\(\lim_{x\rightarrow -1} f(x)\) = ??
\(\lim_{x\rightarrow -1} f(x) = f(-1)\) (How to prove?)
+129852 1. If the function is continuous at x = -1, f(-1) and lim as x → -1 from both sides must have the same value
f(-1) = -1 √ [ - 1 + 3 ] = - √2
And
lim x √ [ x + 3 ] = (-1) √ [-1 + 3] = -√2
x → -1
So this is true
2. Also f(a) must = the limit as it approaches "a" from the right and the left
These are going to look similar
lim x √ [ x + 3 ] = (-1) √ [-1 + 3] = -√2 = f(-1)
x → -1+
And
lim x √ [ x + 3 ] = (-1) √ [-1 + 3] = -√2 = f(-1)
x → -1-
So...we have proved that
1. The limit - from both sides - is the same as the function value at x = -1
2. The limit from the right side is the same as the function value at x = -1
3. The limit from the left side is the same as the function value at x = -1
