Question

In Exercises 15–22, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$ \lim _{x \rightarrow 3}(4-x) $$

Short answer

The limit of \(4 - x\) as \(x\) approaches 3 is 1.

Step-by-step solution

Understand the problem

We want to find the limit of the function \(4 - x\) as \(x\) approaches 3. In simpler terms, we want to find out what value the function approaches as \(x\) gets closer and closer to 3.

Substitute the approach value of \(x\)

We have that, \(x\) is approaching 3, so we substitute \(x\) in the function \(4 - x\) with 3. This gives us \(4 - 3\).

Calculate the result

After substituting, we do the calculation: \(4 - 3 = 1\).

Key concepts

Limit of a Function

The concept of the limit of a function is at the heart of calculus. It's an essential idea that deals with what the output of a function is approaching as its input gets closer and closer to some value. In technical terms, the limit of a function as the variable 'x' approaches a certain value 'a' is the value that the function's output gets arbitrarily close to, provided that 'x' is close enough to 'a' but not equal to 'a'.

In the gradual process of finding this approaching value, we say that 'x' tends toward 'a', and we're interested in the behavior of the function near 'a', not necessarily at 'a' itself. This distinction is important because the function might not even be defined at 'a', or it might have a value different from the limit. For example, if we examine the limit of the function \(4 - x\) as \(x\) approaches 3, we aren't concerned with the function's value at \(x = 3\) itself, but rather what value \(4 - x\) is nearing as \(x\) gets increasingly close to 3.

Approaching a Point Calculation

Calculating how a function behaves as it approaches a specific point is a step-by-step process. The goal is to examine the output of the function when the input value is near a certain point. For instance, in the example \(\lim _{x \rightarrow 3}(4-x)\), 'x' is approaching the value of 3. To find out what happens, we can substitute 'x' with numbers that are extremely close to 3, both less than and greater than 3. The idea is that by getting closer from both sides, if the function approaches the same single number from both directions, then we've found the limit.

We can improve our understanding by considering two scenarios: one where 'x' approaches 3 from the left (values smaller than 3) and one from the right (values larger than 3). If both scenarios yield the same result, then the function has a limit at that point. In this case, by substituting 3 into \(4 - x\), we get the same result from either side, hence, the limit as 'x' approaches 3 is 1.

Calculating Limits from a Graph

To calculate limits from a graph, we focus on the visual behavior of the function's graph as it nears a designated point. This graphical perspective can confirm what we may suspect from the algebraic approach. On a graph, we look at the path the curve takes as we move closer to the point of interest, which is visually portrayed by the 'x' value we're considering.

For the exercise \(\lim _{x \rightarrow 3}(4-x)\), imagine a graph where the y-axis represents the function's output and the x-axis represents the input. As we trace the graph with our eyes from left to right, honing in on the point where 'x' is 3, we observe the curve's direction. Do the left side (approaching from values of 'x' less than 3) and the right side (approaching from values greater than 3) coincide at the same y-value as 'x' approaches 3? If yes, that y-value is our limit. The visualization complements our earlier algebraic calculation and provides a confirmatory visual cue to the limit's value – in this case, the y-value of 1 where the x-value is close to 3.