Discuss the continuity of the function f given by f(x)=∣x∣ at x=0.
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Step-by-Step Solution
Step 1: Define the absolute value function
The absolute value function, f(x)=∣x∣, can be defined piecewise. For non-negative values of x, f(x) is simply x. For negative values of x, f(x) is −x, which makes the output positive.
Step 2: Check the function value at x=0
To check for continuity at x=0, we first need to find the value of the function at this point. Substituting x=0 into the function, we get f(0)=∣0∣, which is 0.
Step 3: Calculate the left-hand limit
Now, we calculate the left-hand limit as x approaches 0. Since x is approaching 0 from the left side, x<0, so we use the definition f(x)=−x. Substituting x=0 into −x gives us 0.
Step 4: Calculate the right-hand limit
Next, we calculate the right-hand limit as x approaches 0. Since x is approaching 0 from the right side, x≥0, so we use the definition f(x)=x. Substituting x=0 into x gives us 0.
Step 5: Compare limits and function value
For a function to be continuous at a point, the left-hand limit, the right-hand limit, and the function value at that point must all be equal. In this case, all three values are 0.