Evaluate the Limit limit as x approaches 4 from the left of 1/(x-4)
The problem presented is asking to determine the behavior of the function 1/(x-4) as the variable x gets closer and closer to the value of 4 from the left side. In other words, you need to find the limit of the function as x approaches the number 4 specifically from values less than 4. This is a question in calculus that deals with limits, which are fundamental in understanding the behavior of functions at specific points, particularly where the function might not be defined or may exhibit some form of discontinuity.
$\underset{x \rightarrow 4^{-}}{lim} \frac{1}{x - 4}$
Answer
Solution:
Step 1:
We observe that the numerator of the function $1/(x-4)$ remains constant at $1$, while the denominator $x - 4$ gets closer to zero as $x$ approaches $4$ from the left side. Since $x$ is less than $4$, the denominator is negative, leading to a negative value for the entire fraction. As the denominator approaches zero, the magnitude of the fraction increases indefinitely in the negative direction. Thus, the limit is $-\infty$.
Solution: "The limit of the function $1/(x-4)$ as $x$ approaches $4$ from the left is $-\infty$, due to the denominator approaching zero from the negative side, causing the value of the function to decrease without bound."
Knowledge Notes:
Limits: The concept of a limit describes the behavior of a function as its argument approaches a particular value or infinity.
One-Sided Limits: A one-sided limit is the value that a function approaches as the independent variable approaches a given point from one side (either from the left or the right).
Infinity in Limits: When dealing with limits, if a function grows without bound as the variable approaches a certain value, the limit is said to be infinite. If the function decreases without bound, the limit is negative infinity.
Behavior Near Vertical Asymptotes: When the denominator of a fraction approaches zero, the function may exhibit a vertical asymptote. The limit at this point is typically infinite or negative infinite, depending on the sign of the function as it approaches the asymptote.
Latex Formatting: In mathematical documents, Latex is used to render formulas and equations with high-quality typography. In this case, $$1/(x-4)$$is used to represent the function in question.
Solution
Step 1:
The limit of the function $1/(x-4)$ as $x$ approaches $4$ from the left is evaluated by considering the signs of the numerator and denominator. The numerator is a constant positive value, and the denominator becomes increasingly negative and close to zero as $x$ approaches $4$ from the left. This causes the value of the function to become a large negative number, leading to a limit of $-\infty$.
