Check the continuity of the function f given by f(x)=2x+3 at x=1.
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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 simpler terms, the limit of f(x) as x approaches a must be equal to the function's value at a.
Step 2: Evaluate the function at x=1
First, we find the value of the function f(x) at the given point x=1. We substitute x=1 into the function's definition f(x)=2x+3 to get f(1)=2(1)+3=5.
Step 3: Evaluate the limit as x approaches 1
Now, we find the limit of the function f(x) as x approaches 1. Since f(x)=2x+3 is a polynomial function, its limit as x approaches any value can be found by direct substitution. So, limx→1(2x+3)=2(1)+3=5.
Step 4: Compare function value and limit
We compare the value of the function at x=1, which is f(1)=5, with the limit of the function as x approaches 1, which is limx→1f(x)=5. Since both values are equal, the function is continuous at x=1.