17–83. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 1 (4 tan⁻¹ x- π) / (x-1)

Accepted Answer

Welcome back, everyone. Evaluate the limit of the given expression. Use laptal's rule if required. Limit as X approaches 2 from the left of inverse of sin of x minus 1 minus pi divided by 2 all divided by X minus 2. We're given 4 answer choices A says -1 B says 0, C 1 and the infinity. Let's go ahead and rewrite the limit limit as x approaches 2 from the left of inverse of sin of x minus 1 minus pi divided by 2 all divided by X minus 2. If we attempt the direct substitution, we're going to get inverts of sin of 2 minus 1, so that's inverts of sine of 1, which gives us pi divided by 2, minus pi divided by 2. Divided by 2 minus 2. So basically this form is 0 divided by 0, which is an indeterminate form that we can use whenever we want to apply Lapito's rules to identify the value of the limit, right? If it's infinity divided by infinity or 0 divided by 0, we're good to go. So now the rule says that we want to differentiate the numerator. So we're going to take the derivative of inverse of sine of X minus 1 minus pi divided by 2. That would be our derivative for the top term. And then we want to differentiate the denominator. So now let's identify the derivatives. We have to understand that the derivative of X minus 2 is 1 based on the power rule, right? The derivative of X is 1, the derivative of -2 is 0. So we're only left with the derivative of the numerator. We get limit as X approaches 2 from the left. What is the derivative of the inverse of sine? Well, we have to recall that it is equal to 1 divided by square root of. 1 minus the square of the argument, so in this case it's x minus 1. Squared and now because our argument is not just x well we need to apply the chain rule and multiply by the derivative of X minus 1 and then we are subtracting the derivative of pi divided by 2, which is simply zero. And now once again the derivative of X minus 1 is 1, right, based on the power rule. So now that we have our derivative, we can attempt direct substitution once again, which gives us limit as x approaches 2 from the left of a 1 divided by square root of 1 minus X minus 1 squad. We're going to take one in the numerator and divide that by square root of. 1 minus 2 minus 1 squad, right, so 2 minus 1 gives us 1 and 1 squared is simply 1. Now notice that x approaches 2 from the left, which means that 2 minus 1 is going to be 1. From the left, right, or simply speaking, this difference will be slightly less than 1 because we're approaching 2 from the left. And therefore, when we square a number that is slightly less than 1, we're still going to get a number that is slightly less than 1, meaning the value under the radical is defined for this limit. So essentially 1 minus 1. From the left gives us 0 from the right. So overall we simply get 1 divided by square root of 0 from the right, which is 1 divided by an infinitely small number, and this is going to be equal to positive infinity, a positive number divided by a number that approaches 0. So by definition this is infinity and therefore our final answer corresponds to the answer choice D infinity. Thank you for watching.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 1 (4 tan⁻¹ x- π) / (x-1)

Key Concepts

Limits

l'Hôpital's Rule

Inverse Tangent Function

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