Question

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 1} f(x), \text { where } f(x)=\left\\{\begin{array}{ll}{x^{3}+1,} & {x<1} \\ {x+1,} & {x \geq 1}\end{array}\right. $$

Short answer

The function has a limit as \(x\) approaches 1 if and only if \(\lim_{x \rightarrow 1^-} f(x) = \lim_{x \rightarrow 1^+} f(x)\). If these limits are not equal, the function has no limit as \(x\) approaches 1.

Step-by-step solution

Calculate the left-hand limit

Calculate the limit of the function as \(x\) approaches 1 from the left side (values less than 1). For this function, this limit is \(\lim_{x \rightarrow 1^-} f(x) = \lim_{x \rightarrow 1^-} (x^3 + 1)\). Substitute \(x = 1\) into the function to find the left-hand limit.

Calculate the right-hand limit

Calculate the limit of the function as \(x\) approaches 1 from the right side (values greater or equal than 1). For this function, this limit is \(\lim_{x \rightarrow 1^+} f(x) = \lim_{x \rightarrow 1^+} (x + 1)\). Substitute \(x = 1\) into the function to find the right-hand limit.

Compare the left-hand and the right-hand limit

The limit as \( x\) approaches 1 of the function exists only if the left-hand limit equals the right-hand limit. If they're equal, then that's the limit; otherwise, the limit does not exist, as the function does not approach a single unique value.

Key concepts

Left-Hand Limit

In calculus, understanding different types of limits is crucial, and one of these is the left-hand limit. This term refers to the value that a function approaches as the input (or x-value) gets closer and closer to a certain point from the left side. It's like sneaking up to a number on a number line from the lower values.

To calculate it, we look at what's happening to our function as we get infinitesimally close to our point of interest, but only from one direction – in this case, the left. Let's take the scenario where we're trying to find the left-hand limit of a function as it approaches the number 1. Formally, we write it as \( \lim_{x \rightarrow 1^-} f(x) \). That little minus sign means we only care about values of x that are less than 1. We substitute values that are slightly less than one into the function and observe how the output behaves. If these values approach a certain number, that's our left-hand limit. Remember, if the function behaves differently on either side of the point, then there may not be an overall limit, but we can still talk about left-hand and right-hand limits separately.

Right-Hand Limit

The right-hand limit is like the mirror image of the left-hand limit. It describes what the function is approaching as the input approaches a specific value from the right, from values that are greater. For a function near the same point, this would be written as \( \lim_{x \rightarrow 1^+} f(x) \). The plus sign indicates we're interested in values of x that are just greater than 1.

To find this, we consider values just a bit larger than our point of interest—again, 1 in this case—and plug those into our function. As with the left-hand limit, if these output values get closer and closer to a single value, that's our right-hand limit. If both left-hand and right-hand limits are equal and finite, the overall limit at that point exists. If they don't match, the function has a jump discontinuity at that point and the overall limit does not exist.

Piecewise Functions

When dealing with piecewise functions, we're essentially looking at a mathematical patchwork quilt. These functions are defined by different expressions in different parts of their domain—a domain being the set of all possible x-values. Think of them as having different rules for different stretches of the x-axis.

For example, in the function provided in the exercise, we have one rule (\(x^3 + 1\)) for x values less than 1, and another rule (\(x + 1\)) for x values that are 1 or greater. They are two distinct expressions sewn together at the point where x equals 1. When looking for limits in piecewise functions, we have to examine each 'piece' separately with regards to the side of the function that we're approaching from. It's vital to keep in mind where the pieces change, as this can be where interesting things like discontinuities can occur.

Limits of Functions

In the grand picture of calculus, the concept of limits of functions is the foundation on which the entire subject rests. A limit seeks to describe the behavior of a function as the input approaches a particular value, without necessarily reaching that value. It's a prediction of where you think the function's output will end up, even if you never actually get there in the function itself.

When finding limits, we're not always guaranteed that they exist. For a limit to exist at a certain point, the function must approach the same value from both the left and the right sides. If it doesn't—if the left-hand and right-hand limits don't match up—then we say the limit at that point doesn't exist. Limits are a powerful tool not just for understanding the immediate behavior of functions but also for deciphering continuity and laying the groundwork for derivatives, which measure change.