Find the limit. lim x → 0 x 3 + 5 x 2 − 7 x + 3 \lim_{x\rarr0}x^3+5x^2-7x+3 lim x → 0 x 3 + 5 x 2 − 7 x + 3

In this problem, we're asked to find the limit of x cubed plus 5x squared minus 7x plus 3 as x approaches 0. So here we can just go ahead and plug 0 into this polynomial in order to get the limit. So this gives me a 0 cubed plus 5 times 0 squared minus 7 times 0 plus 3. Now all of these are going to end up being 0 so they all go away. And I am just left here with 3 as the limit of this function as x approaches 0. Let me know if you have any questions. 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.
$\lim_{x\rarr0}x^3+5x^2-7x+3$

$0$

$2$

$3$

$5$

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the limit of the polynomial function $ x^3 + 5x^2 - 7x + 3 $ as $ x \to 0 $. A limit evaluates the behavior of a function as the input approaches a specific value.

Step 2: Recall the property of limits for polynomials. For a polynomial function, the limit as $ x \to c $ can be found by directly substituting $ x = c $ into the polynomial, provided the function is continuous at that point. Polynomials are continuous everywhere, so substitution is valid here.

Step 3: Substitute $ x = 0 $ into the polynomial $ x^3 + 5x^2 - 7x + 3 $. This means replacing every occurrence of $ x $ in the expression with 0.

Step 4: Simplify the resulting expression. After substitution, calculate each term: $ (0)^3 $, $ 5(0)^2 $, $ -7(0) $, and the constant term $ 3 $. Add these values together to find the limit.

Step 5: Conclude that the limit of the polynomial as $ x \to 0 $ is the value obtained after simplification. This is the final result of the limit.