If 
 
lim 
x→1 
f(x) − 7
x − 1
 = 5
, evaluate 
 
lim 
x→1 
f(x).


We are given the limit:

\[
\lim_{x \to 1} \frac{f(x) - 7}{x - 1} = 5.
\]

This expression is in the form of a difference quotient, which resembles the definition of the derivative of \( f(x) \) at \( x = 1 \). Specifically, we can interpret it as:

\[
f'(1) = 5.
\]

This tells us that the derivative of \( f(x) \) at \( x = 1 \) is 5. However, we are asked to evaluate \( \lim_{x \to 1} f(x) \), which is the value of \( f(x) \) as \( x \) approaches 1.

To evaluate this limit, notice that the term \( \frac{f(x) - 7}{x - 1} \) suggests that \( f(x) \) approaches 7 as \( x \to 1 \). Here's why:

- For the expression \( \frac{f(x) - 7}{x - 1} \) to remain finite (in this case, equal to 5) as \( x \to 1 \), the numerator \( f(x) - 7 \) must approach 0, meaning \( f(x) \) must approach 7.

Thus, we conclude:

\[
\lim_{x \to 1} f(x) = 7.
\]

Let me know if you want more details, or if you have any questions! 

Here are some related questions:
1. What is the definition of a derivative?
2. How can the limit form of the derivative be derived?
3. What are the conditions for a function to be continuous at a point?
4. What is L'Hôpital's Rule, and when is it applicable in limits?
5. How does the limit definition relate to the slope of a tangent line?

**Tip:** In limits involving a difference quotient, you can often interpret the result as a derivative, which provides insights into the behavior of the function at a given point.

If lim x→1 (f(x) − 7) / (x − 1) = 5, evaluate lim x→1 f(x).

Solve the calculus problem where lim x→1 (f(x) − 7) / (x − 1) = 5. This detailed explanation covers the concept of limits, the definition of a derivative, and how to determine the behavior of a function as x approaches 1. Ideal for students in Grades 11-12 studying advanced calculus topics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation