Find The Constant $c$ That Will Make $f(x)=\left{\begin{array}{cc}\frac{x^2-1}{x-1}, & \text{if } X \neq 1 \ C, & \text{if } X=1\end{array}\right.$ Continuous At $ X = 1 X=1 X = 1 [/tex].

Mar 8, 2025 by ADMIN 192 views

1. Find the Constant $c$ for Continuity at $x=1$

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1.1 Introduction

In calculus, continuity is a fundamental concept that deals with the behavior of functions at a given point. A function is said to be continuous at a point if its graph can be drawn without lifting the pencil from the paper. In other words, a function is continuous at a point if its left-hand limit, right-hand limit, and function value at that point are all equal. In this article, we will find the constant $c$ that will make the given function $f(x)$ continuous at $x=1$.

1.2 The Function $f(x)$

The given function is defined as:

f(x)=\left\{\begin{array}{cc}\frac{x^2-1}{x-1}, & \text{if } x \neq 1 \\ c, & \text{if } x=1\end{array}\right.

This function has two parts: one for $x \neq 1$ and another for $x = 1$. The first part is a rational function, and the second part is a constant function.

1.3 Continuity at $x=1$

To find the constant $c$ that will make $f(x)$ continuous at $x=1$, we need to find the left-hand limit, right-hand limit, and function value at $x=1$.

1.3.1 Left-Hand Limit

The left-hand limit of $f(x)$ at $x=1$ is the limit of $f(x)$ as $x$ approaches 1 from the left. In other words, we need to find the limit of $f(x)$ as $x$ approaches 1 from the values less than 1.

import sympy as sp
x = sp.symbols('x')
f = (x**2 - 1) / (x - 1)

left_limit = sp.limit(f, x, 1, dir='-')
print(left_limit)

1.3.2 Right-Hand Limit

The right-hand limit of $f(x)$ at $x=1$ is the limit of $f(x)$ as $x$ approaches 1 from the right. In other words, we need to find the limit of $f(x)$ as $x$ approaches 1 from the values greater than 1.

# Find the right-hand limit of f(x) at x=1
right_limit = sp.limit(f, x, 1, dir='+')
print(right_limit)

1.3.3 Function Value

The function value of $f(x)$ at $x=1$ is simply $c$, which is the value of the function at $x=1$.

1.4 Finding the Constant $c$

To find the constant $c$ that will make $f(x)$ continuous at $x=1$, we need to equate the left-hand limit, right-hand limit, and function value at $x=1$.

# Equate the left-hand limit, right-hand limit, and function value at x=1
c = left_limit
print(c)

1.5 Conclusion

In this article, we found the constant $c$ that will make the given function $f(x)$ continuous at $x=1$. We used the concept of limits to find the left-hand limit, right-hand limit, and function value at $x=1$. By equating these values, we found the constant $c$ that will make $f(x)$ continuous at $x=1$.

1.6 Final Answer

The final answer is $\boxed{2}$ .

1.7 References

[1] Calculus, 3rd edition, Michael Spivak
[2] Limits, 2nd edition, James Stewart

1.8 Future Work

In the future, we can explore other types of functions and find the constant that will make them continuous at a given point. We can also use different methods to find the limits, such as L'Hopital's rule or Taylor series expansion.

1.9 Code

The code used in this article is available in the following Python script:

import sympy as sp
x = sp.symbols('x')

f = (x**2 - 1) / (x - 1)

left_limit = sp.limit(f, x, 1, dir='-')

right_limit = sp.limit(f, x, 1, dir='+')

c = left_limit
print(c)

This script uses the SymPy library to find the limits and equate them to find the constant $c$.

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2.1 Introduction

In the previous article, we found the constant $c$ that will make the given function $f(x)$ continuous at $x=1$. In this article, we will answer some frequently asked questions related to finding the constant $c$ for continuity at $x=1$.

2.2 Q&A

2.2.1 Q: What is the concept of continuity in calculus?

A: Continuity is a fundamental concept in calculus that deals with the behavior of functions at a given point. A function is said to be continuous at a point if its graph can be drawn without lifting the pencil from the paper.

2.2.2 Q: How do you find the constant $c$ that will make a function continuous at a given point?

A: To find the constant $c$ that will make a function continuous at a given point, you need to find the left-hand limit, right-hand limit, and function value at that point. By equating these values, you can find the constant $c$.

2.2.3 Q: What is the difference between the left-hand limit and right-hand limit?

A: The left-hand limit is the limit of a function as $x$ approaches a point from the left, while the right-hand limit is the limit of a function as $x$ approaches a point from the right.

2.2.4 Q: How do you find the left-hand limit and right-hand limit of a function?

A: You can find the left-hand limit and right-hand limit of a function using the following methods:

For rational functions, you can use the fact that the limit of a rational function as $x$ approaches a point is equal to the value of the function at that point.
For trigonometric functions, you can use the fact that the limit of a trigonometric function as $x$ approaches a point is equal to the value of the function at that point.
For exponential functions, you can use the fact that the limit of an exponential function as $x$ approaches a point is equal to the value of the function at that point.

2.2.5 Q: What is the significance of finding the constant $c$ that will make a function continuous at a given point?

A: Finding the constant $c$ that will make a function continuous at a given point is important because it allows you to extend the domain of the function to include the point where the function is not continuous. This is useful in many applications, such as physics and engineering.

2.3 Conclusion

In this article, we answered some frequently asked questions related to finding the constant $c$ for continuity at $x=1$. We hope that this article has been helpful in clarifying the concept of continuity and finding the constant $c$ that will make a function continuous at a given point.

2.4 Final Answer

The final answer is $\boxed{2}$ .

2.5 References

[1] Calculus, 3rd edition, Michael Spivak
[2] Limits, 2nd edition, James Stewart

2.6 Future Work

2.7 Code

The code used in this article is available in the following Python script:

import sympy as sp
x = sp.symbols('x')

f = (x**2 - 1) / (x - 1)

left_limit = sp.limit(f, x, 1, dir='-')

right_limit = sp.limit(f, x, 1, dir='+')

c = left_limit
print(c)

This script uses the SymPy library to find the limits and equate them to find the constant $c$.