Find the limit. lim x → 3 x 2 + 2 x − 3 x − 2 \lim_{x\rarr3}\frac{x^2+2x-3}{x-2} lim x → 3 x − 2 x 2 + 2 x − 3

Here as to find the limit of this irrational function x squared plus 2x minus 3 over x minus 2 as x approaches 3. Now with this rational function, remember that we can only directly sub in our value of x if it doesn't make our denominator 0. Now here since I'm looking at x approaching 3 if I actually plug that value into my function I get 3 squared plus 2 times 3 minus 3 over 3 minus 2. Now 3 minus 2 is going to give me 1 it doesn't give me 0 so I am good to go here. Now if I actually continue actually doing this algebra here, 3 squared is going to give me 9+6minus3 and that's all divided by 1, that 3 minus 2. Now 9+6minus3 will give me 12 and since this is just being divided by 1, that is also my final answer, 12, as the limit of this function as x approaches 3. Let me know if you have any questions here. I'll see you in the next one.

22. Limits & Continuity

Finding Limits Algebraically

Precalculus

NEW 22. Limits & Continuity

Finding Limits Algebraically

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Multiple Choice

Find the limit.
$\lim_{x\rarr3}\frac{x^2+2x-3}{x-2}$

$0$

$3$

$12$

$DNE$

Verified step by step guidance

Step 1: Identify the type of limit problem. This is a rational function limit where direct substitution of x = 3 results in an indeterminate form (0/0).

Step 2: Factor the numerator. The expression x^2 + 2x - 3 can be factored into (x - 1)(x + 3).

Step 3: Simplify the expression. Substitute the factored form into the limit expression: $ \frac{(x - 1)(x + 3)}{x - 2} $.

Step 4: Cancel common factors. Notice that there are no common factors between the numerator and the denominator, so the expression cannot be simplified further.

Step 5: Evaluate the limit using L'Hôpital's Rule. Since direct substitution leads to an indeterminate form, apply L'Hôpital's Rule by differentiating the numerator and the denominator separately and then substituting x = 3.