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.

1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and 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

First, identify the type of limit problem. This is a rational function where both the numerator and the denominator are polynomials.

Check if direct substitution of x = 3 into the function results in an indeterminate form like 0/0. Substitute x = 3 into the numerator and denominator: $ x^2 + 2x - 3 $ becomes $ 3^2 + 2(3) - 3 $ and $ x - 2 $ becomes $ 3 - 2 $.

Calculate the values: The numerator becomes $ 9 + 6 - 3 = 12 $ and the denominator becomes $ 1 $. Since the denominator is not zero, the function is not indeterminate at x = 3.

Since the function is not indeterminate, the limit can be found by direct substitution. Substitute x = 3 into the simplified expression $ \frac{x^2 + 2x - 3}{x - 2} $ to find the limit.

The limit as x approaches 3 is the value of the function at x = 3, which is $ \frac{12}{1} = 12 $. Therefore, the limit is 12.