which expression can be multiplied by the numerator and denominator to help evaluate $lim_{x\rightarrow…

which expression can be multiplied by the numerator and denominator to help evaluate $lim_{x\rightarrow - 1}\frac{sqrt{x - 1}+1}{x + 1}$?\n$x - 1$\n$x + 1$\n$sqrt{x - 1}-1$\n$sqrt{x - 1}+1$

which expression can be multiplied by the numerator and denominator to help evaluate $lim_{x\rightarrow - 1}\frac{sqrt{x - 1}+1}{x + 1}$?\n$x - 1$\n$x + 1$\n$sqrt{x - 1}-1$\n$sqrt{x - 1}+1$

Answer

Explanation:

Step1: Recall the conjugate - rule

To simplify expressions involving square - roots in limits, we use the conjugate. For an expression of the form (a + b), its conjugate is (a - b). Here, the numerator is (\sqrt{x - 1}+1), and its conjugate is (\sqrt{x - 1}-1). When we multiply ((a + b)(a - b)), we get (a^{2}-b^{2}) according to the difference - of - squares formula ((a + b)(a - b)=a^{2}-b^{2}). In the context of limits with square - root expressions, multiplying the numerator and denominator by the conjugate of the numerator helps to rationalize the numerator and simplify the limit.

Step2: Analyze the given limit

We have (\lim_{x\rightarrow - 1}\frac{\sqrt{x - 1}+1}{x + 1}). By multiplying the numerator and denominator by (\sqrt{x - 1}-1), the numerator becomes ((\sqrt{x - 1}+1)(\sqrt{x - 1}-1)=(\sqrt{x - 1})^{2}-1^{2}=x-1 - 1=x - 2) (using the difference - of - squares formula ((a + b)(a - b)=a^{2}-b^{2})). This simplifies the process of evaluating the limit.

Answer:

C. (\sqrt{x - 1}-1)