To evaluate the limit
[tex]\[
\lim_{x \rightarrow 1} \frac{x^2 + 3x - 4}{x - 1},
\][/tex]
we can follow these steps:
1. Substitute the value of [tex]\( x \)[/tex] into the function:
First, we try substituting [tex]\( x = 1 \)[/tex] directly into the function to see what we get:
[tex]\[
\frac{1^2 + 3(1) - 4}{1 - 1} = \frac{1 + 3 - 4}{0} = \frac{0}{0}.
\][/tex]
We end up with an indeterminate form, [tex]\(\frac{0}{0}\)[/tex]. This suggests that we need to simplify the expression further.
2. Factor the numerator:
We need to factor the numerator [tex]\( x^2 + 3x - 4 \)[/tex]. The quadratic expression can be factored as follows:
[tex]\[
x^2 + 3x - 4 = (x + 4)(x - 1).
\][/tex]
3. Simplify the expression:
Substitute the factorization back into the original limit expression:
[tex]\[
\frac{x^2 + 3x - 4}{x - 1} = \frac{(x + 4)(x - 1)}{x - 1}.
\][/tex]
Notice that the [tex]\( x - 1 \)[/tex] terms in the numerator and the denominator cancel out (for [tex]\( x \neq 1 \)[/tex]):
[tex]\[
\frac{(x + 4)(x - 1)}{x - 1} = x + 4.
\][/tex]
4. Evaluate the limit of the simplified expression:
Now that we have simplified the function to [tex]\( x + 4 \)[/tex], we can directly substitute [tex]\( x = 1 \)[/tex]:
[tex]\[
\lim_{x \rightarrow 1} (x + 4) = 1 + 4 = 5.
\][/tex]
Therefore, the limit is
[tex]\[
\boxed{5}.
\][/tex]
