Find the value of f(2) given the function f(x) and its limits.
Understand the Problem
The question involves calculating limits of a function at certain points and determining some properties of that function. It outlines steps for approaching the solution, likely including evaluating the function and its limits as x approaches certain values.
Answer
The function value is $f(2) = 3$ and the limit is $\lim_{x \to 2} f(x) = 4$.
Answer for screen readers
The final results are:
- $f(2) = 3$
- $\lim_{x \to 2} f(x) = 4$
Steps to Solve
- Evaluate the function at x = 2
To find $f(2)$, we look directly at the definition of the function. Since $f(x) = 3$ when $x = 2$, we have: $$ f(2) = 3 $$
- Calculate the limit as x approaches 2
We need to find the limit of $f(x)$ as $x$ approaches 2. Since $f(x) = x + 2$ for $x \neq 2$, we compute: $$ \lim_{x \to 2} f(x) = \lim_{x \to 2} (x + 2) = 2 + 2 = 4 $$
- Compare the limit with the function value
Now we compare the function value at $x = 2$ with the limit we just calculated:
- Since $f(2) = 3$ and $\lim_{x \to 2} f(x) = 4$, this indicates a discontinuity.
The final results are:
- $f(2) = 3$
- $\lim_{x \to 2} f(x) = 4$
More Information
In this case, we have found that while the function $f(x)$ is defined at $x = 2$, it does not match the limiting value as $x$ approaches 2. This illustrates a classic example of a removable discontinuity, where the limit exists but is not equal to the value at the discontinuity.
Tips
- Confusing the value of the function at the point with the limit. Ensure you understand that they can be different, which indicates discontinuity.
- Failing to state that the limit exists and is different from the function value.
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