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

In this problem, we're asked to find the limit of the rational function 3x2+7x divided by x as x approaches 0. Now since this is a rational function, I want to check and see if plugging my value in would make my denominator 0. Now here since my denominator is just x and I'm looking at my limit as x approaches 0, if I were to plug that 0 in, that would make my denominator 0. So that means here we need to go ahead and factor. So let's go ahead and factor this function. Now here we're still trying to find the limit as x approaches 0, but let's go ahead and factor that numerator. Now both of these terms have an x in them. So that means that I can factor out an x in order to get 3x+7. And then that's all divided by that denominator, which cannot be because it is just x. Now here, I can go ahead and cancel out that common factor of x, and I am left to find the limit as x approaches 0 of this function, 3x+7. Now from here, I can just plug that 0 in to get my limit, which will give me 3 times 0+7. 3 times 0 is just 0, so my limit here is just 7. Let me know if you have any questions, and I'll see you in the next video.

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.
$\lim_{x\rarr0}\frac{3x^2+7x}{x}$

$0$

$7$

$-3$

DNE

Verified step by step guidance

Step 1: Recognize that the given problem involves finding the limit of a rational function as x approaches 0. The function is $ \frac{3x^2 + 7x}{x} $.

Step 2: Simplify the expression $ \frac{3x^2 + 7x}{x} $ by factoring out $ x $ from the numerator. This gives $ \frac{x(3x + 7)}{x} $.

Step 3: Cancel the common $ x $ term in the numerator and denominator, leaving $ 3x + 7 $. Note that this simplification is valid for $ x \neq 0 $.

Step 4: Substitute $ x = 0 $ into the simplified expression $ 3x + 7 $. This step evaluates the limit as $ x \to 0 $.

Step 5: Conclude the value of the limit based on the substitution. If the function simplifies to a finite value, that is the limit. If it does not simplify or leads to an undefined result, the limit does not exist (DNE).