To solve the limit [tex]\(\lim_{{x \rightarrow 1}} \frac{x^2 + 3x - 4}{x - 1}\)[/tex] step-by-step, let’s proceed as follows:
1. Identify the expression: We have the rational function [tex]\(\frac{x^2 + 3x - 4}{x - 1}\)[/tex].
2. Simplify the numerator: The numerator is a quadratic expression [tex]\(x^2 + 3x - 4\)[/tex]. We can try to factorize it.
[tex]\[x^2 + 3x - 4 = (x + 4)(x - 1)\][/tex]
3. Rewrite the original limit: Substitute the factored form of the numerator back into the limit expression.
[tex]\[\lim_{{x \rightarrow 1}} \frac{(x + 4)(x - 1)}{x - 1}\][/tex]
4. Cancel common factors: Notice that [tex]\((x - 1)\)[/tex] appears in both the numerator and the denominator, so we can cancel this term because we are interested in the limit as [tex]\(x\)[/tex] approaches 1, not exactly at [tex]\(x = 1\)[/tex].
[tex]\[\lim_{{x \rightarrow 1}} (x + 4)\][/tex]
5. Evaluate the limit: Now, we can directly substitute [tex]\(x = 1\)[/tex] into the simplified expression.
[tex]\[x + 4\][/tex]
[tex]\[\lim_{{x \rightarrow 1}} (x + 4) = 1 + 4 = 5\][/tex]
Therefore, the limit is:
[tex]\[
\lim_{{x \rightarrow 1}} \frac{x^2 + 3x - 4}{x - 1} = 5
\][/tex]
Thus, the solution to the limit is [tex]\(5\)[/tex].
