Examine whether the function f given by f(x)=x2 is continuous at x=0.
Get the complete, step-by-step math solution for: "Examine whether the function f given by f(x)=x^{2} is continuous at x=0.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Understand Continuity Condition
For a function f(x) to be continuous at a point x=a, three conditions must be met: the function must be defined at a, the limit of the function as x approaches a must exist, and these two values must be equal. In this problem, we need to check continuity at x=0.
Step 2: Evaluate the function at x=0
First, we evaluate the function f(x)=x2 at the given point x=0. Substituting x=0 into the function, we find that f(0) is 02, which equals 0.
Step 3: Evaluate the limit as x approaches 0
Next, we need to find the limit of the function f(x)=x2 as x approaches 0. Since f(x) is a polynomial function, its limit as x approaches any point can be found by direct substitution.
Step 4: Calculate the limit
By substituting x=0 into the expression x2, we find that the limit of f(x) as x approaches 0 is 02, which is 0.
Step 5: Compare function value and limit
We found that f(0)=0 and limx→0f(x)=0. Since these two values are equal, the condition for continuity is satisfied.