Use l’Hôpital’s rule to find the limits in Exercises 7–52.
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Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

22. lim (x → 1) (x - 1) / (ln x - sin πx)

Accepted Answer

Welcome back everyone. Use Lapito's rule to find the limit. Limit as X approaches 1 of X minus LNX minus 1 divided by X minus 1 squared. Let's begin this problem by rewriting the limit. Our function is X minus LN of X minus 1 divided by. X minus 1 squared. Remember that the first step is to attempt direct substitution, so we got one. 11 of 1 minus 1. Divided by 1 minus 1 squared. LN of 1 is equal to 0, so we get 1 minus 1 in the numerator, which gives 0. In the denominator, we get 1 minus 1 squared, which is also 0. 0 divided by 0 is an indeterminate form that is suitable for laptose rules to be applied. So we can proceed with the actual application of the rule, right? According to the rule, we want to take the derivative of the numerator. And we want to take the derivative of them. The nominator. So now let's find each derivative. We're going to get limit as X approaches 1 of the derivative of X is 1. Minus the derivative of LN X is 1 divided by X, and the derivative of -1 is 0 because it's a constant. In the denominator, we have a composite function. First of all, we're differentiating square with respect to X minus 1, using the power rule, which gives us 2 multiplied by X minus 1. And using the chain rule we're multiplying by the derivative of the inner function, so the derivative of X minus 1. Remember that the derivative of X minus 1 is equal to 0 because the derivative of X is 1 and the derivative of -1 is equal to 0. It's a constant. So we can just say that we now have 2 multiplied by X minus 1 in the denominator. Once again, what we're going to do from here is perform direct substitution to see if we get a finite value. So we get limit as X approaches one. Of 1 minus 1 divided by X, right? We can actually drop the limit because we are attempting the right substitution at this stage, and we can show that it's 1 minus 1 divided by 1 in the numerator, and in the denominator, we get 2 multiplied by 1 minus 1. Again, this is 0 in the numerator. And 0 in the denominator, which essentially means that we want to apply the rule once again. So now let's differentiate the numerator. 1 minus 1 divided by X derivative, and divided by 2 multiplied by X minus 1 derivative. We can factor have the constant for simplicity. We get limit as X approaches 1. The derivative of 1 is 0. We can then factor out the negative sign, right? And we want to differentiate a 1 divided by X, which is X is the power of -1. So here we can apply the power rule. We can bring down the exponent. We get negative X to the power of negative 2. And X to the power of -2 can be written as a 1 divided by X2. Additionally, we can now cancel out the two negative signs. And in the denominator we simply get 2, we know that the derivative of X minus 1 is 1. Now we can combine two with x2d because we have that double division and again let's attempt direct substitution for the new function. We get 1 divided by 2 multiplied by 1 squared, which is equal to 1/2, and that's Now our final limit. Thank you for watching.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.22. lim (x → 1) (x - 1) / (ln x - sin πx)

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Use l’Hôpital’s rule to find the limits in Exercises 7–52.

22. lim (x → 1) (x - 1) / (ln x - sin πx)