Problem 55
Question
If $$\lim _{x \rightarrow 1} \frac{f(x)-8}{x-1}=10,$$ find $$\lim _{x \rightarrow 1} f(x).$$
Step-by-Step Solution
Verified Answer
\( \lim_{x \rightarrow 1} f(x) = 8 \).
1Step 1: Understanding the Limit Equation
We are given the equation \( \lim_{x \rightarrow 1} \frac{f(x)-8}{x-1} = 10 \). This represents the derivative of a function at \( x = 1 \), specifically since it's in the format of a difference quotient for a continuous function \( f(x) \). The quotient indicates that \( x \) approaches 1 as \( f(x) \) approaches 8.
2Step 2: Applying L'Hôpital's Rule
To solve the limit \( \lim_{x \rightarrow 1} \frac{f(x)-8}{x-1} \), if setting this function equals 10, indicates the limit of the difference quotient which equals the derivative of \( f(x) \) at \( x = 1 \). But to find \( \lim _{x \rightarrow 1} f(x) \), we directly substitute \( x = 1 \).
3Step 3: Finding \( \lim _{x \rightarrow 1} f(x) \)
Given that the limit implies \( f(x) \) approaches 8 as \( x \) approaches 1, the expression \( f(x) - 8 \) must approach 0 as \( x \) approaches 1. Therefore, \( \lim_{x \rightarrow 1} f(x) = 8 \).
Key Concepts
LimitsDerivativesL'Hôpital's RuleDifference Quotient
Limits
Limits are fundamental building blocks in calculus in understanding behavior around specific points in a function. They help us explore what happens to a function's value as the input approaches a particular number. For instance, in the exercise where we have \( \lim_{x \rightarrow 1} \frac{f(x)-8}{x-1} = 10 \), this limit directs us towards understanding behavior of \( f(x) \) near \( x = 1 \).
The limit indicates how \( f(x) \) behaves very close to \( x = 1 \), even if we do not substitute \( x = 1 \) directly into \( f(x) \). It's like asking "What value do we get closer to if we get closer to 1?" as opposed to "What value do we have exactly at 1?" This concept of predicting behavior, without directly measuring, is essential in calculus.
The limit indicates how \( f(x) \) behaves very close to \( x = 1 \), even if we do not substitute \( x = 1 \) directly into \( f(x) \). It's like asking "What value do we get closer to if we get closer to 1?" as opposed to "What value do we have exactly at 1?" This concept of predicting behavior, without directly measuring, is essential in calculus.
- Limits show potential value trends of functions.
- They can help identify continuous behavior or predict future outcomes in a function.
- Calculating limits helps us in many areas such as finding slopes of tangents.
Derivatives
The derivative of a function is a measure of how a function's output changes as its input changes. It's like asking how fast something is moving if we know how far it moved in a certain amount of time. In the context of our exercise, the limit \( \lim_{x \rightarrow 1} \frac{f(x)-8}{x-1} = 10 \) actually represents the derivative of \( f(x) \) at \( x = 1 \).
This is because this limit is in the form of the difference quotient, which we often use to define the derivative, specifically as \( x \) approaches a particular value. The derivative provides us with the slope of the tangent line to the function at a certain point, showing how steeply the function rises or falls.
This is because this limit is in the form of the difference quotient, which we often use to define the derivative, specifically as \( x \) approaches a particular value. The derivative provides us with the slope of the tangent line to the function at a certain point, showing how steeply the function rises or falls.
- Derivatives help understand rate of change and slopes of curves.
- They can explain how physical phenomena like velocity and acceleration work.
- In calculus, they're crucial for finding maximum or minimum values of functions.
L'Hôpital's Rule
L'Hôpital's Rule is a powerful tool in calculus for determining limits that initially appear as indeterminate forms, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). It suggests that under certain conditions, the limit of a quotient of two functions can be found by taking the limit of the quotient of their derivatives. This greatly simplifies the solution process when directly substituting does not work.
In our exercise, although it was not directly needed, recognizing that \( \lim_{x \rightarrow 1} \frac{f(x)-8}{x-1} = 10 \) can resemble a situation where L'Hôpital's Rule might be applied is important, as it indicates that the function has a derivative at that point. In dealing with indeterminate forms, L'Hôpital's Rule offers a structured approach to unravel tricky limits.
In our exercise, although it was not directly needed, recognizing that \( \lim_{x \rightarrow 1} \frac{f(x)-8}{x-1} = 10 \) can resemble a situation where L'Hôpital's Rule might be applied is important, as it indicates that the function has a derivative at that point. In dealing with indeterminate forms, L'Hôpital's Rule offers a structured approach to unravel tricky limits.
- Useful for resolving 0/0 or \(\infty/\infty\) forms.
- Relies on differentiation to simplify complex limit problems.
- Offers a systematic method instead of random guesswork.
Difference Quotient
The difference quotient is an expression used in calculus to define the derivative. It takes the form \( \frac{f(x+h) - f(x)}{h} \) as \( h \) approaches 0 and is fundamental in understanding changes over intervals. In the exercise, we observe \( \lim_{x \rightarrow 1} \frac{f(x)-8}{x-1} = 10 \); this is a closed form of the difference quotient, typically used in calculations to find a derivative.
This quotient shows how the change in the output \( f(x) \) compares to the change in input \( x \). It's like computing how much a function changes from one point to the next and is a key step in finding the function's differentiation, helping to grasp the concept of instantaneous rate of change.
This quotient shows how the change in the output \( f(x) \) compares to the change in input \( x \). It's like computing how much a function changes from one point to the next and is a key step in finding the function's differentiation, helping to grasp the concept of instantaneous rate of change.
- Forms the basis of understanding derivatives with finite differences.
- Enables visualization of how a function evolves over intervals.
- Introduces the concept of slope for a curve, paving the way for calculus.
Other exercises in this chapter
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