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Chapter 4: Problem 91

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow 1^{+}}\left(\frac{1}{x-1}-\frac{1}{\sqrt{x-1}}\right)$$

Short Answer

Expert verified
Answer: The limit of the given expression as x approaches 1 from the positive side is 1.

Step by step solution

01

Rewrite the given expression

To evaluate this limit, the first step is to rewrite the given expression. We have: $$\lim _{x \rightarrow 1^{+}}\left(\frac{1}{x-1}-\frac{1}{\sqrt{x-1}}\right)$$
02

Find a common denominator for the two fractions

To combine the fractions, we need a common denominator. In this case, it will be the product of the two denominators, that is, \((x-1)\sqrt{x-1}\). So, let's rewrite the expression with the common denominator. $$\lim _{x \rightarrow 1^{+}}\left(\frac{1}{(x-1)\sqrt{x-1}}\left((\sqrt{x-1} - 1)\right)\right)$$
03

Apply the conjugate

We notice that we have a difference of squares in the numerator with \(\sqrt{x-1}\) and 1. We can use the conjugate to simplify the expression further. The conjugate of the numerator is \((\sqrt{x-1} + 1)\). Multiply both numerator and denominator by the conjugate and then simplify the expression: $$\lim _{x \rightarrow 1^{+}}\left(\frac{(\sqrt{x-1} - 1)(\sqrt{x-1} + 1)}{(x-1)\sqrt{x-1}(\sqrt{x-1} + 1)}\right)$$
04

Simplify the expression

After multiplying the conjugate, we can simplify the expression. In the numerator, we have a difference of squares, and in the denominator, we can cancel out one term of \(\sqrt{x-1}\): $$\lim _{x \rightarrow 1^{+}}\left(\frac{(x-1)}{(x-1)(\sqrt{x-1} + 1)}\right)$$ Now, we can cancel out \((x - 1)\) from both the numerator and denominator: $$\lim _{x \rightarrow 1^{+}}\left(\frac{1}{\sqrt{x-1} + 1}\right)$$
05

Evaluate the limit

Finally, we can evaluate the limit as x approaches 1 from the positive side: $$\lim _{x \rightarrow 1^{+}}\left(\frac{1}{\sqrt{x-1} + 1}\right) = \frac{1}{\sqrt{1-1} + 1} = \frac{1}{0+1} = 1$$ Therefore, the limit of the given expression as x approaches 1 from the positive side is 1.

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Most popular questions from this chapter

Evaluate one of the limits l'H么pital used in his own textbook in about 1700: \(\lim _{x \rightarrow a} \frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}}{a-\sqrt[4]{a x^{3}}},\) where \(a\) is a real number.

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