Question

True or False If lim \(f ( x ) = 2\) and \(\lim f ( x ) = 2 ,\) then $$\lim _ { x \rightarrow c } f ( x ) = 2 .$$ Justify your answer.

Short answer

The statement is true.

Step-by-step solution

Understanding the problem

In the problem, it's mentioned that the limit of the function \(f(x)\) is 2 as \(x\) approaches any value. By the basic limit definition, if for each \(x\) in the domain of a function \(f(x)\), the function \(f(x)\) approaches a certain number as \(x\) gets infinitesimally close to a value \(c\), then that number is the limit of the function at \(c\). This should apply to any specific value \(c\).

Applying the limit concept

As per the limit concept, if \(\lim f ( x ) = 2\) for all \(x\), it also implies that the limit as \(x\) approaches any specific value \(c\) should also be equal to 2. More formally, if \(\lim _ { x \rightarrow a } f ( x ) = 2\) for all \(a\) in the domain of function \(f(x)\), then \(\lim _ { x \rightarrow c } f ( x )\), for a specific \(c\) in domain of function \(f(x)\), should also be equal to 2.

Finalizing the answer

So, the statement is true. The behavior of the function at the limit value \(c\), or even whether \(f(c)\) is defined, does not affect what the limit of the function as \(x\) approaches \(c\) is.

Key concepts

Calculus

Calculus is a field of mathematics that studies continuous change, and it is a foundational component for many scientific disciplines. It's generally divided into two branches: differential calculus, which concerns the concept of a derivative, and integral calculus, which concerns the concept of an integral.

At its core, calculus deals with understanding behavior of functions and how these functions model real-world phenomena. The essence of calculus lies in the concepts of limits and the instantaneous rate of change. For example, when we are trying to find the speed of a car at a precise moment, we are using the principles of differential calculus to determine this instantaneous rate of change.

Continuity in Functions

Continuity in functions is a concept that describes how smoothly a function behaves. A function is said to be continuous at a point if there is no abrupt change in its value at that point. This is formally expressed by saying that a function f(x) is continuous at a point c if the limit of f(x) as x approaches c is equal to the value of the function at c, that is, f(c).

Intuitive Understanding of Continuity:

Imagine drawing the graph of a function without lifting your pencil from the paper. If you can do this over the entire domain of the function, then the function is said to be continuous. Discontinuities can occur in the form of jumps, holes, or vertical asymptotes in the graph of the function.

Limit Definition

The limit of a function is a fundamental concept in calculus that deals with the behavior of that function as its argument approaches a certain value. Formally, we say that the limit of f(x) as x approaches a is L (written as \(\lim_{x \rightarrow a} f(x) = L\)) if we can make the value of f(x) as close as we want to L by taking x sufficiently close to a, but not equal to a.

Significance of Limits:

Limits help us understand the behavior of functions at points that may not be easily evaluated by direct substitution due to discontinuities or indeterminate forms. Limits are also crucial in defining the derivative of a function, which measures the rate of change, and the integral of a function, which measures the accumulation of quantities.

Approaching a Limit

Approaching a limit involves getting closer to a point on the function's domain where we want to evaluate the function's limit. This concept is all about what happens to the function's value as the input value gets infinitely close to some number c, even if the function is not defined at c.

How to Approach a Limit:

When working with limits, we consider both the left-hand limit (as x approaches c from the left) and the right-hand limit (as x approaches c from the right). In order for the limit to exist, both these one-sided limits must be equal.

It's important to understand that limits describe what is happening around the point, not necessarily at the point. This is why limits can exist even at points where the function itself may be undefined or at points of discontinuity, making them a powerful tool in analyzing the behavior of functions.