Find the limit by creating a table of values. lim x → 2 3 x 2 + 5 x + 1 \lim_{x\rarr2}3x^2+5x+1 lim x → 2 3 x 2 + 5 x + 1

Here, we're asked to find the limit by creating a table of values. And the limit that we wanna find here is the limit of 3 x squared plus 5 x plus 1 as x approaches 2. Now in creating a table of values here, we're gonna look at values that are really, really close to 2, and we're gonna start with values that are actually less than 2 and getting really close to it. Now when we think of these values, you can also think of sort of closing in on this 2. So we think of values like 1.99 and getting even closer to 21.999. Now actually plugging these values into my function here, 3x squared plus 5x plus 1. Doing that for that 1.99 here, this would give me 3 times 1.99 squared plus 5 times 1.99 plus 1. Now actually doing this algebra and getting this value, I will end up getting a value of 22.83 for this value. Now if I plug in that 1.999, that's actually going to give me a value of 22.98, having done that same thing plugging in that value for x in our function. Now it looks like here we're closing in on values that are getting really close to a value of 23 for our function as x approaches 2. But let's also check out our values that are greater than 2 coming in from the other side. Remember that we want to look at values really close to 2, kind of closing in on it from either side. So here, we want to look at values like 2.01 and getting even closer to 0.001. And here, plugging these values into our function, if I plug in this 2.001, that's gonna give me 23.01. And if I plug in this 2.01, that's gonna give me 23.17. So again, here on this side, I see that we're kind of closing in on a value of 23 for our function. So that tells me that my limit of 3x squared plus 5x plus 1 as x approaches 2 is equal to 23. Thanks for watching, and let me know if you have any questions.

1. Limits and Continuity

Introduction to Limits

Business Calculus

1. Limits and Continuity

Introduction to Limits

Struggling with Business Calculus?

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Multiple Choice

Find the limit by creating a table of values.
$\lim_{x\rarr2}3x^2+5x+1$

$1$

$10$

$23$

$21$

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the limit of the function $ f(x) = 3x^2 + 5x + 1 $ as $ x $ approaches 2. This means we want to determine the value that $ f(x) $ gets closer to as $ x $ gets closer to 2.

Step 2: Create a table of values. Choose values of $ x $ that are close to 2 from both the left (e.g., 1.9, 1.99, 1.999) and the right (e.g., 2.1, 2.01, 2.001). For each $ x $, substitute it into the function $ f(x) = 3x^2 + 5x + 1 $ to calculate the corresponding $ f(x) $ values.

Step 3: Substitute the chosen $ x $-values into the function. For example, calculate $ f(1.9) = 3(1.9)^2 + 5(1.9) + 1 $, $ f(1.99) = 3(1.99)^2 + 5(1.99) + 1 $, and so on. Similarly, calculate $ f(2.1) $, $ f(2.01) $, and $ f(2.001) $.

Step 4: Observe the trend in the $ f(x) $ values. As $ x $ gets closer to 2 from both sides, the $ f(x) $ values should approach a specific number. This number is the limit of the function as $ x $ approaches 2.

Step 5: Conclude the limit. Based on the trend observed in the table of values, determine the value that $ f(x) $ approaches as $ x $ gets closer to 2. This is the value of $ \lim_{x \to 2} 3x^2 + 5x + 1 $.