Question

Assume \(\lim _{x \rightarrow 3} g(x)=4\) and \(f(x)=g(x)\) whenever \(x \neq 3 .\) Evalu\(\lim _{x \rightarrow 3} f(x),\) if possible.

Short answer

Answer: The limit of f(x) as x approaches 3 is 4, if possible.

Step-by-step solution

Recall the definition of the limit

The limit of a function as x approaches a particular value is the value the function approaches when x gets infinitely close to that value, but not necessarily equals the function's value at that point.

Evaluate limit for g(x)

We are given that the limit of g(x) as x approaches 3 is 4, which can be written as: \(\lim_{x \rightarrow 3} g(x) = 4\)

Evaluate limit for f(x)

Since f(x) and g(x) are equal for all values of x except 3, and g(x) has a limit when x approaches 3, we expect f(x) to have the same limit: \(\lim_{x \rightarrow 3} f(x) = \lim_{x \rightarrow 3} g(x)\)

Determine the limit of f(x) as x approaches 3

Based on our observation in step 3 and the fact that the limit of g(x) when x approaches 3 is 4, we can determine the limit of f(x) as x approaches 3: \(\lim_{x \rightarrow 3} f(x) = 4\) Therefore, the limit of f(x) as x approaches 3 is 4, if possible.

Key concepts

Limit of a Function

The concept of the limit of a function is a fundamental part of calculus that deals with understanding the behavior of functions as they approach a certain point. In simpler terms, a limit asks the question: What value does the function get close to as the input (or x) gets close to a particular number?

For instance, if we have \(\lim_{x \rightarrow a} f(x) = L\), this notation signifies that as x approaches the number a, the function f(x) approaches the value L. It's important to note that the function doesn't need to be exactly L when x is a; it's all about what value f(x) is approaching as x comes infinitely close to a.

If we examine our exercise, the limit of g(x) as x approaches 3 is given to be 4. Symbolically, this is \(\lim_{x \rightarrow 3} g(x) = 4\). This implies that no matter how close we get to x being 3, g(x) approaches the value of 4. This tells us about the predictable behavior of the function around that specific point, even if we don't know the function's value at the point.

Approaching a Point

Discussing approaching a point in the context of limits involves considering the values of a function as the input values get nearer and nearer to a specific number, without necessarily reaching that number. The behavior during this approach reveals what the function is doing right around the point of interest.

In the mathematical exercise, the input x is approaching the value of 3. We must consider what happens to f(x) and g(x) as x gets very close to 3, but is not exactly 3. This concept is critical because it differentiates between the limit and the actual value at a point, which may or may not be the same. \(\lim_{x \rightarrow 3} f(x) = 4\) does not mean f(3) equals 4, but rather that if you made a list of values for f(x) where x is slightly less than 3 and slightly more than 3, and you looked at what those f(x) values were getting closer to, you'd find they are getting closer to 4.

Function Behavior Near a Point

Continuity and Discontinuity

When we talk about function behavior near a point, we are looking at how the function behaves on either side of that point. If a function approaches the same value from both the left side and the right side as it nears a particular point, we can say the function is continuous at that point. However, if the function approaches different values, it is discontinuous or has a jump at that point.

In our example, we imply a sort of continuity as we are treating f(x) and g(x) as having the same behavior around x = 3. However, we are given that f(x) does not necessarily equal g(x) at x = 3, suggesting there may be a point of discontinuity. Even so, if the limits as x approaches 3 from both sides are equal (which is assumed but not explicitly given in our problem), we can say that f(x) has a limit as x approaches 3. This hypothetical smooth behavior near the point allows for robust predictions about the function's output in that neighborhood without needing the exact value at the point.