Problem 37

Question

37-40. For each piecewise linear function, find: a. $\lim _{x \rightarrow 4^{-}} f(x)$ b. $\lim _{x \rightarrow 4^{+}} f(x)$ c. $\lim _{x \rightarrow 4} f(x)$ $$ f(x)=\left\\{\begin{array}{ll} 3-x & \text { if } x \leq 4 \\ 10-2 x & \text { if } x>4 \end{array}\right. $$

Step-by-Step Solution

Verified

Answer

a) -1, b) 2, c) does not exist.

1Step 1: Understand the function

The piecewise function is defined: For $x \leq 4$, $f(x) = 3-x$;For $x > 4$, $f(x) = 10-2x$.We need to find the limits as $x$ approaches 4 from the left, right, and both directions.

2Step 2: Limit from the left

To find $\lim_{x \to 4^-} f(x)$, we consider the function when $x \leq 4$: \[f(x) = 3 - x\]At $x = 4$, $f(x) = 3 - 4 = -1$. So, $\lim_{x \to 4^-} f(x) = -1$.

3Step 3: Limit from the right

To find $\lim_{x \to 4^+} f(x)$, we consider the function when $x > 4$: \[f(x) = 10 - 2x\]As $x$ approaches 4 from the right:\[f(4) = 10 - 2(4) = 2\]So, $\lim_{x \to 4^+} f(x) = 2$.

4Step 4: Calculate the two-sided limit

To find $\lim_{x \to 4} f(x)$, both one-sided limits must equal each other. Since $\lim_{x \to 4^-} f(x) = -1$ and $\lim_{x \to 4^+} f(x) = 2$, these limits are not equal.Thus, $\lim_{x \to 4} f(x)$ does not exist.

Key Concepts

Limits of FunctionsOne-Sided LimitsContinuous Functions

Limits of Functions

Understanding the limits of a function helps us predict the behavior of the function as the input approaches a particular value. It is a fundamental concept in calculus.

For a limit to exist at a point, the function must approach the same value from all directions as it gets closer to that specific point. When dealing with a piecewise function, it is important to analyze each piece as the input approaches different values.

Given the function:

For $x \leq 4$, $f(x) = 3 - x$
For $x > 4$, $f(x) = 10 - 2x$

while finding $ \lim_{x \to 4} f(x) $, it is necessary to find both $ \lim_{x \to 4^-} f(x) $ and $ \lim_{x \to 4^+} f(x) $ to determine if they converge to the same value.

One-Sided Limits

One-sided limits focus only on the behavior of a function as it approaches a given point from one side, either the left or the right.

Limit from the Left ($\lim_{x \to 4^-} f(x)$)

When considering a limit from the left, we use the part of the function that describes the values just below the point of interest. This means we look at $f(x) = 3-x$ for $x \leq 4$. As $x$ approaches 4 from values less than 4, substitute $x$ with 4:

$f(4) = 3 - 4 = -1$

Thus, $\lim_{x \to 4^-} f(x) = -1$.

Limit from the Right ($\lim_{x \to 4^+} f(x)$)

If we want the limit from the right, we focus on the function defined for $x > 4$, which is $f(x) = 10 - 2x$. Approaching 4 from the right gives:

$f(4) = 10 - 2(4) = 2$

So, $\lim_{x \to 4^+} f(x) = 2$.

Continuous Functions

A function is continuous at a point if it is smoothly connected there, meaning there are no breaks, jumps, or holes at the point. Mathematically, for a function to be continuous at $x = c$, the following must hold:

$f(c)$ is defined
$\lim_{x \to c} f(x)$ exists
$\lim_{x \to c} f(x) = f(c)$

For the piecewise function provided, we must check the two one-sided limits. Since $\lim_{x \to 4^-} f(x) = -1$ and $\lim_{x \to 4^+} f(x) = 2$, these values do not equal each other. Thus, $\lim_{x \to 4} f(x)$ does not exist, indicating a jump discontinuity at $x = 4$.

This jump discontinuity tells us that the function is not continuous at $x = 4$, because the left and right limits are not the same, resulting in a distinct gap in the graph of the function.

Problem 37