Chapter 14: Q. 7 (page 905)

find the limit $\lim_{x \to 1} \frac{x - 1}{x^{3} - 1}$

Short Answer

Expert verified

$\lim_{x \to 1} \frac{x - 1}{x^{3} - 1} = \frac{1}{3}$

Step by step solution

Step 1. Given information

$\lim_{x \to 1} \frac{x - 1}{x^{3} - 1}$

Step 2. Domain of the function

We can find limit of rational function by $\lim_{x \to c} f (x) = f (c)$ if c is in the domain of f(x) .So to find the domain of f(x)= $\frac{x - 1}{x^{3} - 1}$ we have to put denominator =0

$x^{3} - 1 = 0 x = 1$

So the domain of f(x) is all real number except 1.

Step 3. limit of the function

As c=1 in our limit so we have to simplify the f(x).

$\lim_{x \to 1} \frac{x - 1}{x^{3} - 1} = \lim_{x \to 1} \frac{x - 1}{(x - 1) (x^{2} + x + 1)} = \lim_{x \to 1} \frac{1}{x^{2} + x + 1} = \frac{1}{(1)^{2} + (1) + 1} = \frac{1}{3}$

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find the limit $\lim_{x \to 1} \frac{x - 1}{x^{3} - 1}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Domain of the function

Step 3. limit of the function

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find the limit $\lim_{x \to 1} \frac{x - 1}{x^{3} - 1}$