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Evaluate: lim(xrarr2) (x^(3)-2x^(2))/(x^...

Evaluate: `lim_(xrarr2) (x^(3)-2x^(2))/(x^(2)-5x+6)`

A

`4`

B

`-4`

C

`-1/4`

D

`1/4`

Text Solution

AI Generated Solution

The correct Answer is:
To evaluate the limit \( \lim_{x \to 2} \frac{x^3 - 2x^2}{x^2 - 5x + 6} \), we will follow these steps: ### Step 1: Substitute \( x = 2 \) First, we will substitute \( x = 2 \) directly into the expression to check if we can evaluate the limit directly. \[ \text{Numerator: } 2^3 - 2 \cdot 2^2 = 8 - 8 = 0 \] \[ \text{Denominator: } 2^2 - 5 \cdot 2 + 6 = 4 - 10 + 6 = 0 \] Since both the numerator and denominator evaluate to 0, we have an indeterminate form \( \frac{0}{0} \). We need to simplify the expression. ### Step 2: Factor the numerator and denominator Next, we will factor both the numerator and the denominator. **Numerator:** \[ x^3 - 2x^2 = x^2(x - 2) \] **Denominator:** To factor the denominator \( x^2 - 5x + 6 \), we look for two numbers that multiply to 6 and add to -5. These numbers are -2 and -3. \[ x^2 - 5x + 6 = (x - 2)(x - 3) \] ### Step 3: Rewrite the limit Now we can rewrite the limit using the factored forms: \[ \lim_{x \to 2} \frac{x^2(x - 2)}{(x - 2)(x - 3)} \] ### Step 4: Cancel common factors We can cancel the common factor \( (x - 2) \) from the numerator and the denominator (valid since we are taking the limit as \( x \) approaches 2, not equal to 2): \[ \lim_{x \to 2} \frac{x^2}{x - 3} \] ### Step 5: Substitute \( x = 2 \) again Now we substitute \( x = 2 \) into the simplified expression: \[ \frac{2^2}{2 - 3} = \frac{4}{-1} = -4 \] ### Final Answer Thus, the limit is: \[ \lim_{x \to 2} \frac{x^3 - 2x^2}{x^2 - 5x + 6} = -4 \] ---

To evaluate the limit \( \lim_{x \to 2} \frac{x^3 - 2x^2}{x^2 - 5x + 6} \), we will follow these steps: ### Step 1: Substitute \( x = 2 \) First, we will substitute \( x = 2 \) directly into the expression to check if we can evaluate the limit directly. \[ \text{Numerator: } 2^3 - 2 \cdot 2^2 = 8 - 8 = 0 ...
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NAGEEN PRAKASHAN-LIMITS AND DERIVATIVES-Miscellaneous Exercise
  1. Evaluate: lim(xrarr2) (x^(3)-2x^(2))/(x^(2)-5x+6)

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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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  7. 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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  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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