Sketch the graph of a function with a jump discontinuity at x = 1. Then, evaluate the limits from the left, right, overall, and the value of the function you graphed at x = 1. Use proper notation.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
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![### Educational Website Content
#### Mathematical Analysis and Continuity
Welcome to our section on mathematical analysis and continuity. In this segment, we will explore concepts like jump discontinuities and the continuity of piecewise functions.
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### Problem 4: Jump Discontinuity
**Task:**
Sketch the graph of a function that experiences a jump discontinuity at \( x = 1 \). After sketching, evaluate the limits from the left, the right, the overall limit, and the value of the function at \( x = 1 \), using proper mathematical notation.
**Steps:**
1. **Sketching the Graph:**
- A jump discontinuity at \( x = 1 \) indicates that the function has different left-hand and right-hand limits at that point.
- You'll draw two separate curves on either side of \( x = 1 \), ensuring they do not meet at \( x = 1 \).
2. **Evaluating the Limits:**
- **Left-hand limit (\( \lim_{x \to 1^-} f(x) \))**: Determine the value the function approaches as \( x \) approaches 1 from the left.
- **Right-hand limit (\( \lim_{x \to 1^+} f(x) \))**: Determine the value the function approaches as \( x \) approaches 1 from the right.
- **Overall limit (\( \lim_{x \to 1} f(x) \))**: Since there is a jump discontinuity, the left-hand and right-hand limits will not be equal, thus the overall limit does not exist.
- **Function value at \( x = 1 \) (\( f(1) \))**: Check the defined value of the function at \( x = 1 \).
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### Problem 5: Continuity of a Piecewise Function
**Given Function:**
\[
h(x) =
\begin{cases}
\frac{x^2 + 2x - 5}{x + 7} & \text{if } x \neq -3 \\
8 & \text{if } x = -3
\end{cases}
\]
**Task:**
Discuss the continuity of the function \( h(x) \) at \( x = -3 \) and justify your answer using the concept of limits.
**Steps:**
1. **Function Analysis:**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadfa6f6f-7ef4-405b-8188-183f47fb167b%2Fa1363210-6729-4736-9b37-aec29c33d570%2Fst9s69y.jpeg&w=3840&q=75)
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