Calculator limits Use a calculator to approximate the followin...

Question

Calculator limits Use a calculator to approximate the following limits.

lim x🠂0 e^3x-1 / x

Accepted Answer

Welcome back, everyone. Approximate the limit using a calculator. Limit as X approaches 0 of E to the power of 11 X minus 1 divided by X. We're given 4 answer choices A says 0, B1, C 11, and D 22. So if we want to approximate the limit, we have to recall the limit existence theorem. First of all, we have to understand that the limit exists if we have a limit as X approaches A from the right of some function F of X, and that limit must be equal to the limit of F of X as X approaches A from the left, right? So those two limits from the left and from the right must be equal to each other, then we can conclude that the limit exists as X approaches that point A itself of F of X. So, what we're going to do is simply understand that in this case, X approaches 0, right? So what we want to do is simply check. Specific x values where x approaches 0 from the left and X approaches 0 from the right. So we're going to build a table with x values in our function, which is the power of 11 X minus 1 divided by X, meaning we're going to take. Several X values that are close to 0, and we're going to evaluate the function by substituting those X values into the function, right? So, let's begin. We can take negative 0.001, so we're taking a very small negative value close to 0, checking what happens as X approaches 0 from the left. And now if we approach 0, well, we can add an extra 0 to check the other values such as -0.0001, followed by an additional one such as negative 0.00001. So, as we can see, we're getting closer and closer to zero from the left, and now if we want to check those limits from the right, we can essentially take positive 0.01. Or let's begin with 0.001 to have symmetrical values, followed by 0.0001 and 0.00001. And now let's value the value of the function for each of those x values. So if we substitute X equals -0.001, we're going to get 10.93972. That's the approximate value, right? We're going to use five decimal places. Now, considering the next X value -0.0001, we're going to get approximately 10.99395. For our next value we're going to get approximately 10.99940 and now considering the positive values, if we begin with 0.001, we're going to get approximately 11.06072 followed by X equals 0.0001, which gives us approximately 11.00605, and for our final value we're going to get 11.00061. So now what can we see? Well, when we are approaching. 0 from the left. Our function approaches 11, right? We have 10.93, 10.993, 10.999. So we get closer and closer to 11. Now, if we are approaching 0 from the right. In order for the limit to exist, we must be approaching the same value of the 11, and that's the case. We have 11.06, 11.006, and 11.0006, right? So as we get closer and closer to zero from the right, our function gets closer and closer to 11. So those two values agree, which means that the limit. As x approaches 0 of E to the power of 11 x minus 1 divided by X exists, these two values are equal and it is equal to 11. So our final answer to this problem corresponds to the answer choice C. Thank you for watching.

Calculator limits Use a calculator to approximate the following limits.lim x🠂0 e^3x-1 / x

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Calculator limits Use a calculator to approximate the following limits.
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