Find all values of c such that the limit exists

lim
x-->1 (x^2 + 3x + c) / (x-1)

Well, to find the values of c for which the limit exists, we need to consider when the expression (x^2 + 3x + c)/(x-1) is defined.

Let's take a closer look at the denominator, (x-1). The limit will not exist if (x-1) approaches 0, as division by zero is undefined. Hence, we need to find the values of c for which (x-1) is not equal to zero.

To avoid division by zero, we need to have (x - 1) ≠ 0, which gives us x ≠ 1. Thus, the limit exists for all values of c as long as x is not equal to 1.

To find the values of c such that the limit exists, we need to determine if the expression (x^2 + 3x + c) / (x-1) can be simplified to remove the discontinuity at x = 1.

Let's start by evaluating the expression for x = 1.

(x^2 + 3x + c) / (x-1) = (1^2 + 3(1) + c) / (1-1)
= (1 + 3 + c) / 0

Since division by zero is undefined, this expression is undefined when x = 1.

To remove the discontinuity at x = 1, we need to simplify the expression so that the denominator is not zero.

Factoring the numerator:
(x^2 + 3x + c) = (x + )(x + )

If we want to remove the discontinuity at x = 1, then (x - 1) must be a factor of the numerator.

So, let's set (x - 1) as a factor of (x + )(x + ):

(x - 1)(x + ) = x^2 + x - x - 1 = x^2 + (x - x) - 1 = x^2 - 1

Comparing this to the original numerator, we have:

x^2 + 3x + c = x^2 - 1

By comparing the corresponding coefficients, we can determine the value of c.

Matching the coefficients of the terms:
3x = 0x --> 3 = 0 (Since the coefficients of x are not equal, this equation is not possible.)

Therefore, there is no value of c that makes the limit exist in this case. The limit is undefined.

To find the values of c such that the limit exists, we need to analyze the expression (x^2 + 3x + c) / (x-1) as x approaches 1.

First, let's simplify the expression by factoring the numerator if possible:

x^2 + 3x + c = (x + ?) (x + ?)

We need to find two numbers that multiply to c and add up to 3. These numbers are related to the coefficients of x in the quadratic expression.

The expression (x^2 + 3x + c) / (x-1) is undefined if x - 1 equals 0, which means x = 1. When x = 1, the denominator becomes 0, resulting in division by zero. This is an example of a potential issue that would cause the limit to not exist.

To prevent division by zero, we need to factor out (x - 1) from the numerator so that it cancels out with the denominator:

(x^2 + 3x + c) = (x - 1)(x + ?) + ?

Expanding the right side:

(x - 1)(x + ?) + ? = x^2 + ?x - x + ?? + ? = x^2 + (2 + ?)x + (?? + ?)

We can see that the term multiplying x in the numerator is (2 + ?). For the limit to exist, we need to ensure that the numerator approaches a finite value as x approaches 1 from both sides.

So, for the limit to exist, we need (2 + ?)x + (?? + ?) to approach a finite value as x approaches 1.

This means that (2 + ?) should be equal to 0, so:

2 + ? = 0
? = -2

To summarize, the values of c such that the limit exists are c = -2.

for the limit to exist, x-1 has to divide evenly into

x^2 + 3x + c

let f(x) = x^2 + 3x + c
f(1) = 0 = 1 + 3 + c
c = -4

check:
lim (x^2 + 3x - 4)/(x-1) as x -->1
= lim (x+4)(x-1)/(x-1) as x ---> 1
= lim x+4
= 5 , the limit exists when c = -4