Understanding Discontinuity in the Function f(x) 1 / (x - 2) at x 2

The function ( f(x) frac{1}{x - 2} ) is a classic example to explore discontinuity in mathematical analysis, particularly when ( x 2 ). This article delves into the nature of this discontinuity, illustrating why and how the function behaves at ( x 2 ).

What is a Discontinuous Function?

A function ( f(x) ) is considered discontinuous at a point ( x a ) if any of the following conditions hold:

The function is undefined at ( x a ). The limit of ( f(x) ) as ( x ) approaches ( a ) does not exist. The limit of ( f(x) ) as ( x ) approaches ( a ) exists, but the function value ( f(a) ) does not equal the limit.

In this case, we are examining the function ( f(x) frac{1}{x - 2} ) at ( x 2 ).

Why is f(x) Discontinuous at x 2?

The function ( f(x) frac{1}{x - 2} ) becomes undefined at ( x 2 ) because the denominator is zero when ( x 2 ). This causes the function to approach infinity as ( x ) approaches 2 from either side. Let's illustrate this mathematically:

1. Undefined at x 2:

When ( x 2 ), the denominator ( x - 2 0 ), making ( f(x) frac{1}{0} ) which is undefined.

Vertical Asymptote at x 2

The function ( f(x) ) has a vertical asymptote at ( x 2 ). This means the graph of ( f(x) ) approaches a particular value (in this case, positive or negative infinity) as ( x ) gets arbitrarily close to 2. This is evident on the graph of ( f(x) ), where the line ( x 2 ) is not crossed but approached infinitely closely.

Behavior Near x 2

Let's look at the behavior of ( f(x) ) as ( x ) approaches 2 from the left and from the right:

Left Hand Limit (LHL) as x approaches 2:

As ( x ) approaches 2 from the left, ( x - 2 ) becomes a small negative number. Consequently, ( frac{1}{x - 2} ) becomes a large negative number. Mathematically:

Right Hand Limit (RHL) as x approaches 2:

As ( x ) approaches 2 from the right, ( x - 2 ) becomes a small positive number. Consequently, ( frac{1}{x - 2} ) becomes a large positive number. Mathematically:

Since the left-hand limit and the right-hand limit are not equal, the overall limit does not exist at ( x 2 ).

Graphical Representation

The graphical representation of ( f(x) frac{1}{x - 2} ) with ( x 2 ) will show a vertical asymptote at ( x 2 ). The function will approach positive infinity from the right and negative infinity from the left. This can be visualized as follows:

Conclusion

In summary, the function ( f(x) frac{1}{x - 2} ) is discontinuous at ( x 2 ) because the function is undefined at this point, and the behavior of the function as ( x ) approaches 2 results in an infinite limit from both sides. This discontinuity is a clear violation of the conditions for continuity, making ( x 2 ) a point of discontinuity for the function.

Related Keywords

Discontinuity

Vertical Asymptote

Undefined Function

Limits

Graphing