given that f(x) = sqrt x/x determine the existnece of f(x) = 0 and the limits as x->0 from below and above. thus, determine the continuity of the function at x=0
To determine the existence of f(x) = 0, we need to find the value of f(x) when x = 0.
f(0) = sqrt(0)/0 = 0/0
Since division by zero is undefined, f(0) does not exist.
Next, let's find the limits as x approaches 0 from below and above:
Limit as x approaches 0 from below:
lim x->0- sqrt(x)/x
= lim x->0- 1/sqrt(x)
= 1/0- = -∞
Limit as x approaches 0 from above:
lim x->0+ sqrt(x)/x
= lim x->0+ 1/sqrt(x)
= 1/0+ = +∞
Since the limit as x approaches 0 from below and above do not equal each other or exist, the function is not continuous at x=0.