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

Business Calculus

1. Limits and Continuity

Finding Limits Algebraically

Struggling with Business Calculus?

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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: Recognize that the given problem involves finding the limit of a rational function as x approaches 3. The function is $ \frac{x^2 + 2x - 3}{x - 2} $.

Step 2: Check if direct substitution of $ x = 3 $ into the function results in an indeterminate form (e.g., $ \frac{0}{0} $). Substitute $ x = 3 $ into the numerator $ x^2 + 2x - 3 $ and denominator $ x - 2 $.

Step 3: If direct substitution results in an indeterminate form, factorize the numerator $ x^2 + 2x - 3 $. Look for two numbers that multiply to $-3$ and add to $2$. The factorization is $ (x + 3)(x - 1) $.

Step 4: Rewrite the function as $ \frac{(x + 3)(x - 1)}{x - 2} $. Since $ x - 2 $ does not cancel with any term in the numerator, proceed to evaluate the limit by substituting $ x = 3 $ into the simplified expression.

Step 5: Substitute $ x = 3 $ into the simplified expression $ \frac{(x + 3)(x - 1)}{x - 2} $ to compute the limit. If the denominator becomes zero, the limit does not exist (DNE). Otherwise, calculate the value of the limit.