Question

Explain in your own words, without using \(\varepsilon-\delta\) formality, why \(\lim _{x \rightarrow c} b=b\)

Short answer

The limit is \(b\) because a constant function remains \(b\) for all \(x\).

Step-by-step solution

Understanding the Concept of Limits

The concept of a limit deals with what happens to the value of a function as its input approaches some point. For the limit \(\lim _{x \rightarrow c} b = b\), we are considering a constant function, where the output value is always \(b\), regardless of the input \(x\).

Evaluating the Constant Function

Since the function always returns \(b\) for any input \(x\), when \(x\) approaches \(c\), it doesn’t affect the output value. Thus, the function value at \((c)\) remains \(b\).

Recognizing Continuity

A constant function is continuous everywhere because its value does not change, irrespective of the input. Thus, whether \(x\) approaches \(c\) from the left or right, the function output stays constant at \(b\).

Concluding the Limit

Since the function does not change its value, the limit value as \(x\) approaches any point \(c\) has no room to be anything other than \(b\). Hence, \(\lim _{x \rightarrow c} b = b\).

Key concepts

Understanding Limits

Limits help us understand how a function behaves as its input gets closer and closer to a specific value, called a limit point. Imagine driving towards a destination. Even if you never actually reach it, being able to predict how close you'll get to your goal is like calculating a limit.
For the function where \(lim _{x \rightarrow c} b = b\), we're dealing with a function that has the same output no matter the input. As you approach the point \(c\) on the graph, the output doesn't waver—it sticks to \(b\).

• Limits tell us the anticipated value at a point, even if we're not physically there.
• They help differentiate between sudden changes and stable outcomes in functions.

Constant Function

A constant function is incredibly simple yet quite powerful. No matter what input you feed into the function, the output remains the same. Think of it like always hearing the same sound on a radio station, no matter how far you drive.
In mathematical terms, if you have a function \(f(x) = b\), the output is always \(b\), regardless of the value of \(x\). This unwavering nature is central to understanding why the limit of a constant function remains constant.

• Constant functions do not vary; their graphs are flat horizontal lines.
• Evaluating them is straightforward, as there's no change in output to account for.

Continuity

Continuity is about smoothness in mathematics. A function is continuous if you can draw its graph without lifting your pencil. For a function to be continuous at a certain point, its limit at that point must equal its actual value there.
A constant function is a perfect example of continuity because its output never changes. Whether you're approaching from the left or right, the output remains constant. Think of it like crossing a bridge with no bumps. It's always smooth.

• Continuity ensures no sudden jumps or dips in a function's graph.
• Constant functions meet this seamless transition perfectly as they never vary in outcome.