lim
x-1
3x^2 + 2x - 3
2x - 1

Q: lim x->1 3x^2 + 2x - 3 2x - 1

To solve the limit $\lim_{x \to 1} \frac{3x^2 + 2x - 3}{2x - 1}$, we can try to factor the numerator and see if it simplifies with the denominator. 1. **Factor the numerator**: $3x^2 + 2x - 3$. We look for two numbers that multiply to $3 \times (-3) = -9$ and add to $2$. The numbers are $3$ and $-1$. Rewrite the middle term: \[ 3x^2 + 3x - x - 3 \] Factor by grouping: \[ 3x(x + 1) - 1(x + 1) \] Factor out the common factor: \[ (3x - 1)(x + 1) \] 2. **Simplify the expression**: \[ \frac{(3x - 1)(x + 1)}{2x - 1} \] Since $2x - 1$ does not cancel with any factor in the numerator, we substitute $x = 1$ directly into the simplified expression. 3. **Substitute $x = 1$**: \[ \frac{(3(1) - 1)(1 + 1)}{2(1) - 1} = \frac{(3 - 1)(2)}{2 - 1} = \frac{2 \cdot 2}{1} = 4 \] Thus, the limit is $\boxed{4}$.

Question

lim
x->1
3x^2 + 2x - 3
2x - 1

Accepted Answer

To solve the limit $\lim_{x \to 1} \frac{3x^2 + 2x - 3}{2x - 1}$, we can try to factor the numerator and see if it simplifies with the denominator.

1. **Factor the numerator**: $3x^2 + 2x - 3$.

We look for two numbers that multiply to $3 \times (-3) = -9$ and add to $2$. The numbers are $3$ and $-1$.

Rewrite the middle term:
   $$
   3x^2 + 3x - x - 3
   $$

Factor by grouping:
   $$
   3x(x + 1) - 1(x + 1)
   $$

Factor out the common factor:
   $$
   (3x - 1)(x + 1)
   $$

2. **Simplify the expression**:
   $$
   \frac{(3x - 1)(x + 1)}{2x - 1}
   $$

Since $2x - 1$ does not cancel with any factor in the numerator, we substitute $x = 1$ directly into the simplified expression.

3. **Substitute $x = 1$**:
   $$
   \frac{(3(1) - 1)(1 + 1)}{2(1) - 1} = \frac{(3 - 1)(2)}{2 - 1} = \frac{2 \cdot 2}{1} = 4
   $$

Thus, the limit is $\boxed{4}$.