Check the continuity of the function $f$ given by $f(x)=2 x+3$ at $x=1$.

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, $\lim_{x \to 1} (2x+3) = 2(1)+3 = 5$.

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 $\lim_{x \to 1} f(x) = 5$. Since both values are equal, the function is continuous at $x=1$.

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