Determine the following limits.
lim x→3 1/ x − 3(1 /√x +...

Question

Determine the following limits.

lim x→3 1/ x − 3(1 /√x + 1 − 1/2)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the limit. The limit as X approaches 4. Of 1 divided by X minus 4, all multiplied by 1 divided by the square root of X + 5 minus 1/3. Awesome. So it appears what we're ultimately trying to do, what we're trying to solve for our end goal, is we're trying to evaluate the specific limit. So now that we know that we're evaluating this limit, let's read off our multiple choice answers to see what our final answer might be. A is 127, B is -148, C is -136, and D is -154th. Awesome. So, first off, in order to solve this problem, we need to first recall and recognize the fact that direct substitution will result in a 1 divided by 0 case for the fraction of 1 divided by X minus 4. So once again, we need to note that direct substitution will result in 1 divided by 0, or the fraction of 1 divided by X minus 4. So now that we've acknowledged that fact, we now need to simplify our function. So we need to take 1 divided by X minus 4 and multiply it by 1 divided by the square root of X + 5. And we need to take this and we need to subtract 1/3, so 1 divided by the square root of X + 5 minus 1/3. And we once again we need to simplify the expression, so we need to set this equal to 1 divided by X minus 4, multiplied by 1 divided by the square root of X + 5, so X + 5. Multiplied by 3 divided by 3 minus 1 divided by 3, multiplied by the square root of X + 5 divided by. X + 5. Fantastic. So moving right along here, and don't forget that there's a parenthesis, it got a little bit cut off, but there's a parenthesis there at the very end. So now, in an effort to save time, ultimately we're taking our expression and we're trying to get it down to its most simple form using algebra. So when we get it into our most simple form, which I'm going to skip a few steps. Its most simplified form will mean that it's equal to negative, 1 divided by 3 multiplied by the square root of X + 5. Multiplied by 3 plus the square root of X plus 5. Fantastic, and that is our most simplified expression. So once again, this is our simplified expression is equal to -1 divided by 3 multiplied by the square root of X + 5, all multiplied by 3 plus the square root of X + 5. Awesome. So now our next step is we need to evaluate the limit by applying and recalling and applying direct substitution onto our modified function or into our modified function. So when we do that, when we use direct substitution, we'll be able to write that the limit, which is as X approaches for, so the limit as X approaches 4 of negative. 1 divided by 3 multiplied by the square root of X plus 5, multiplied by 3 plus the square root of X + 5. Fantastic. And this is all equal to when we apply our direct substitution method, -1 divided by 3, so when you take -1 divided by 3. Multiplied by, and we're taking, so we're plugging in our value for 4 because we're evaluating the limit as X approaches 4, so for every value of X, we're plugging in 4. So we're gonna be taking 3 multiplied by this the square root of X plus 5, so X is 4, so when you take 4 + 5. And then we need to multiply it by 3 plus the square root of 4 + 5. Fantastic. So when we plug that into our calculator, we will find, and when we so when we plug it into our calculator, we will find that it's simply equal to -1 divided by 54 or -154, and that is our final answer. Hooray, we did it, and this corresponds to multiple choice answer letter D. Which once again is -154. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Determine the following limits.
lim x→3 1/ x − 3(1 /√x + 1 − 1/2)

Key Concepts

Limits

Indeterminate Forms

Rational Functions

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