Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 20
In Exercises 15鈥�22, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$ \lim _{x \rightarrow 5} \frac{2}{x-5} $$
Short Answer
Expert verified
The limit \( \lim _{x \rightarrow 5} \frac{2}{x-5} \) does not exist.
Step by step solution
01
Evaluate the Limit from the Left Side
Evaluate the limit of the function as \( x \) approaches 5 from the left, formally written as \( \lim _{x \rightarrow 5^-} \frac{2}{x-5} \). As \( x \) approaches 5 from values less than 5, \( (x-5) \) approaches 0 from the negative side, thus the value of \( \frac{2}{x-5} \) becomes increasingly large negative. We can conclude that \( \lim _{x \rightarrow 5^-} \frac{2}{x-5} = -\infty \) .
02
Evaluate the Limit from the Right Side
Next, evaluate the limit of the function as \( x \) approaches 5 from the right, formally written as \( \lim _{x \rightarrow 5^+} \frac{2}{x-5} \). Now, as \( x \) approaches 5 from values greater than 5, \( (x-5) \) approaches 0 from the positive side, and thus the value of \( \frac{2}{x-5} \) becomes increasingly large positive. We can conclude that \( \lim _{x \rightarrow 5^+} \frac{2}{x-5} = \infty \) .
03
Determine if the Limit exists
A limit only exists if the left and right limits match. In this case, since \( \lim _{x \rightarrow 5^-} \frac{2}{x-5} \neq \lim _{x \rightarrow 5^+} \frac{2}{x-5} \), we can conclude that the limit \( \lim _{x \rightarrow 5} \frac{2}{x-5} \) does not exist.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
One-Sided Limits
A one-sided limit refers to the value that a function approaches as the input approaches a particular point from one side, either from the left (left-hand limit) or from the right (right-hand limit).
For the given function \( \frac{2}{x-5} \), we evaluate what happens as the variable \( x \) draws closer to the number 5 from either direction. When approaching from the left (\( x \rightarrow 5^- \)), the function value descends toward negative infinity (\( -\infty \)), signaling that the left-hand limit diverges. Conversely, from the right (\( x \rightarrow 5^+ \)), it soars to positive infinity (\( \infty \)), indicating the right-hand limit also diverges, but in the opposite direction.
For the given function \( \frac{2}{x-5} \), we evaluate what happens as the variable \( x \) draws closer to the number 5 from either direction. When approaching from the left (\( x \rightarrow 5^- \)), the function value descends toward negative infinity (\( -\infty \)), signaling that the left-hand limit diverges. Conversely, from the right (\( x \rightarrow 5^+ \)), it soars to positive infinity (\( \infty \)), indicating the right-hand limit also diverges, but in the opposite direction.
- The presence of different one-sided limits often indicates a discontinuity at that point, implying the overall limit does not exist.
- Understanding one-sided limits is essential when assessing functions' behavior near points of potential discontinuity.
Asymptotic Behavior
The asymptotic behavior of a function describes how the function behaves as the input either gets very large (\( x \rightarrow \infty \)) or decreases without bound (\( x \rightarrow -\infty \)), or as it approaches a particular point where the function becomes unbounded.
In our example, as \( x \) approaches 5, the function \( \frac{2}{x-5} \) does not approach a finite value but instead increases or decreases without bound. This is the hallmark of a vertical asymptote at \( x = 5 \), a line that the graph of the function gets infinitely close to but never actually touches or crosses.
In our example, as \( x \) approaches 5, the function \( \frac{2}{x-5} \) does not approach a finite value but instead increases or decreases without bound. This is the hallmark of a vertical asymptote at \( x = 5 \), a line that the graph of the function gets infinitely close to but never actually touches or crosses.
- A vertical asymptote represents a barrier that the function can approach but not intersect.
- Recognizing asymptotic behavior aids in graphing functions and understanding their long-term trends.
Limits and Infinity
When we talk about limits and infinity, it often involves situations where the function grows without bound as the input approaches a particular value, or as the input itself increases or decreases without limit.
Our function \( \frac{2}{x-5} \) exemplifies this as it has no finite limit at \( x = 5 \); instead, it diverges to positive or negative infinity. It's crucial to differentiate this behavior from a function that simply takes on a very large value; in the case of limits involving infinity, the function doesn't settle at any particular number but continues to grow or decrease endlessly.
Our function \( \frac{2}{x-5} \) exemplifies this as it has no finite limit at \( x = 5 \); instead, it diverges to positive or negative infinity. It's crucial to differentiate this behavior from a function that simply takes on a very large value; in the case of limits involving infinity, the function doesn't settle at any particular number but continues to grow or decrease endlessly.
- This concept is pivotal in calculus as it examines the end behavior of functions and how they react to extreme inputs or approach critical points.
- Expressions like \( \lim_{x \rightarrow a} f(x) = \infty \) do not imply that infinity is a number that the function reaches, but rather that it does not have a bound as \( x \) nears \( a \).
Discontinuity
The concept of discontinuity arises when a function is not continuous at a point: it may have a gap, a jump, or approach different values from each side.
In the context of our exercise, the difference in behavior of \( \frac{2}{x-5} \) when approaching from the left and the right indicates a discontinuity at \( x = 5 \). This specific type of discontinuity, where the function goes to positive or negative infinity, is a point of infinite discontinuity.
In the context of our exercise, the difference in behavior of \( \frac{2}{x-5} \) when approaching from the left and the right indicates a discontinuity at \( x = 5 \). This specific type of discontinuity, where the function goes to positive or negative infinity, is a point of infinite discontinuity.
- Discontinuities are significant in understanding the domain and range of functions, their integrability, and potential for optimization problems.
- While some discontinuities can be fixed by redefining the function at certain points, an infinite discontinuity like the one observed cannot be remedied in this manner.
