Show that the function f given by f(x)={x3+3,1, if x=0 if x=0 is not continuous at x=0.
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Step-by-Step Solution
Step 1: Recall the definition of continuity
For a function 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 this limit must be equal to the function's value at a. In this problem, we need to check these conditions at x=0.
Step 2: Find the function value at x=0
From the given definition of the function, when x=0, the function f(x) is defined as 1. So, f(0)=1.
Step 3: Find the limit of the function as x approaches 0
To find the limit as x approaches 0, we consider the part of the function definition for x=0, which is f(x)=x3+3. We substitute x=0 into this expression to evaluate the limit.
Step 4: Evaluate the limit
Substituting x=0 into the expression x3+3, we get 03+3=0+3=3. Therefore, the limit of f(x) as x approaches 0 is 3.
Step 5: Compare the limit with the function value
We found that the limit of f(x) as x approaches 0 is 3, but the function value at x=0 is 1. Since these two values are not equal, the condition for continuity is not met.