To find the limit of the function [tex]\(\frac{x^2 + 2x - 3}{x^2 + 3x - 4}\)[/tex] as [tex]\(x\)[/tex] approaches 1, we'll proceed with the following steps:
1. Factorize the Numerator and the Denominator:
Let's start by factoring both the numerator [tex]\(x^2 + 2x - 3\)[/tex] and the denominator [tex]\(x^2 + 3x - 4\)[/tex].
- Numerator: [tex]\(x^2 + 2x - 3\)[/tex]
We look for two numbers that multiply to [tex]\(-3\)[/tex] (the constant term) and add up to [tex]\(2\)[/tex] (the coefficient of the linear term).
Factors of [tex]\(-3\)[/tex]: [tex]\(3\)[/tex] and [tex]\(-1\)[/tex]
Thus, [tex]\(x^2 + 2x - 3 = (x + 3)(x - 1)\)[/tex].
- Denominator: [tex]\(x^2 + 3x - 4\)[/tex]
We look for two numbers that multiply to [tex]\(-4\)[/tex] (the constant term) and add up to [tex]\(3\)[/tex] (the coefficient of the linear term).
Factors of [tex]\(-4\)[/tex]: [tex]\(4\)[/tex] and [tex]\(-1\)[/tex]
Thus, [tex]\(x^2 + 3x - 4 = (x + 4)(x - 1)\)[/tex].
2. Rewrite the Function Using the Factorizations:
Using these factorizations, we can rewrite the function as:
[tex]\[
\frac{x^2 + 2x - 3}{x^2 + 3x - 4} = \frac{(x + 3)(x - 1)}{(x + 4)(x - 1)}
\][/tex]
3. Simplify the Expression:
Notice that [tex]\((x - 1)\)[/tex] is a common factor in both the numerator and the denominator. We can cancel out this common factor, provided [tex]\(x \neq 1\)[/tex]:
[tex]\[
\frac{(x + 3)(x - 1)}{(x + 4)(x - 1)} = \frac{x + 3}{x + 4} \quad (\text{for } x \neq 1)
\][/tex]
4. Evaluate the Limit of the Simplified Expression:
Now, we can directly substitute [tex]\(x = 1\)[/tex] into the simplified expression to find the limit:
[tex]\[
\lim_{x \to 1} \frac{x + 3}{x + 4} = \frac{1 + 3}{1 + 4} = \frac{4}{5}
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{\frac{4}{5}}
\][/tex]
