If lim x→0 f(x)=3 and lim x→0 g(x)=1, then find lim x→0 (f(x)+g(x))^2
a) 40
b) -4
c) 16
d) 28
lim (f(x)+g(x))^2 = (lim f(x) + lim g(x))^2 = (3+1)^2 = 16
To find the limit as x approaches 0 of the expression (f(x) + g(x))^2, we can use limit properties and basic algebraic manipulations.
1. Start by finding the limits of the individual functions f(x) and g(x) as x approaches 0:
lim x→0 f(x) = 3
lim x→0 g(x) = 1
2. Next, substitute these limits into the expression (f(x) + g(x))^2:
lim x→0 (f(x) + g(x))^2 = (lim x→0 (f(x) + g(x)))^2
3. Use the limit property that states if the limits of two functions exist, then the limit of their sum exists and is equal to the sum of the limits:
lim x→0 (f(x) + g(x))^2 = (lim x→0 f(x) + lim x→0 g(x))^2
4. Substitute the known limits of f(x) and g(x) into the equation:
lim x→0 (f(x) + g(x))^2 = (3 + 1)^2
5. Simplify the expression:
lim x→0 (f(x) + g(x))^2 = 4^2
6. Evaluate the expression:
lim x→0 (f(x) + g(x))^2 = 16
Therefore, the correct answer is c) 16.