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Evaluate: lim(xrarr1) ((2)/(x^(2)-1)+(1)...

Evaluate: `lim_(xrarr1) ((2)/(x^(2)-1)+(1)/(1-x))`

A

`1/2`

B

`1/4`

C

`-1/2`

D

`-1/4`

Text Solution

AI Generated Solution

The correct Answer is:
To evaluate the limit \[ \lim_{x \to 1} \left( \frac{2}{x^2 - 1} + \frac{1}{1 - x} \right), \] we can start by simplifying the expression. ### Step 1: Factor the denominator The term \(x^2 - 1\) can be factored as: \[ x^2 - 1 = (x - 1)(x + 1). \] So we rewrite the limit as: \[ \lim_{x \to 1} \left( \frac{2}{(x - 1)(x + 1)} + \frac{1}{1 - x} \right). \] ### Step 2: Rewrite the second term Notice that \(1 - x = -(x - 1)\). Therefore, we can rewrite the second term: \[ \frac{1}{1 - x} = -\frac{1}{x - 1}. \] Now the limit becomes: \[ \lim_{x \to 1} \left( \frac{2}{(x - 1)(x + 1)} - \frac{1}{x - 1} \right). \] ### Step 3: Combine the fractions To combine the two fractions, we need a common denominator, which is \((x - 1)(x + 1)\): \[ \frac{2}{(x - 1)(x + 1)} - \frac{1(x + 1)}{(x - 1)(x + 1)} = \frac{2 - (x + 1)}{(x - 1)(x + 1)}. \] This simplifies to: \[ \frac{2 - x - 1}{(x - 1)(x + 1)} = \frac{1 - x}{(x - 1)(x + 1)}. \] ### Step 4: Simplify the expression Notice that \(1 - x = -(x - 1)\), so we can rewrite the expression as: \[ \frac{-(x - 1)}{(x - 1)(x + 1)} = \frac{-1}{x + 1} \quad \text{(for } x \neq 1\text{)}. \] ### Step 5: Evaluate the limit Now we can evaluate the limit as \(x\) approaches 1: \[ \lim_{x \to 1} \frac{-1}{x + 1} = \frac{-1}{1 + 1} = \frac{-1}{2}. \] Thus, the final result is: \[ \boxed{-\frac{1}{2}}. \]

To evaluate the limit \[ \lim_{x \to 1} \left( \frac{2}{x^2 - 1} + \frac{1}{1 - x} \right), \] we can start by simplifying the expression. ...
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NAGEEN PRAKASHAN-LIMITS AND DERIVATIVES-Miscellaneous Exercise
  1. Evaluate: lim(xrarr1) ((2)/(x^(2)-1)+(1)/(1-x))

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  2. Find the derivative of the following functions from first principles:...

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  3. Find the derivative of the following functions (it is to be understand...

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  4. Find the derivative of the following functions (it is to be understand...

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  5. Find the derivative of the following functions (it is to be understand...

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  6. Find the derivative of the following functions (it is to be understand...

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  7. Find the derivative of the following functions (it is to be understand...

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  8. Find the derivative of the following functions (it is to be understand...

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  9. Find the derivative of the following functions (it is to be understand...

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  10. Find the derivative of the following functions (it is to be understand...

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  11. Find the derivative of the following functions (it is to be understand...

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  12. Find the derivative of the following functions (it is to be understand...

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  15. Find the derivative of the following functions (it is to be understand...

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  16. Find the derivative of the following functions (it is to be understand...

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  17. Find the derivative of the following functions (it is to be understand...

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  19. Find the derivative of the following functions (it is to be understand...

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  20. Find the derivative of the following functions (it is to be understand...

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  21. Find the derivative of the following functions (it is to be understand...

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