88–91. Limits Use l’Hôpital’s Rule to evaluate the following l...

Question

88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.

lim x → ∞ (1 − coth x) / (1 − tanh x)

Accepted Answer

Welcome back, everyone. Evaluate the limit using Lapital's rule. Limit as X approaches infinity of 1 minus 1 X divided by 1 minus hyperbolic cotangent X A 1 B 0 C 11/2, and D1. For this problem, let's rewrite our limit. Limit as x approaches infinity. We have 1 minus 1 X and the numerator, and our denominator contains 1 minus hyperbolic cotangent of X. If we attempt direct substitution. And if X approaches infinity, then we understand that tang is going to approach 1. And hyperbolic cotangent is going to approach one as well, right, as X approaches infinity. So we're going to get a 1 minus 1 in the numerator. And 1 minus 1 in the denominator, so that's 0 divided by 0, which is an indeterminate form. And whenever we have this form, we can apply Lapis rule directly. We're going to differentiate our numerator and denominator. Let's begin by performing the differentiation of the numerator. The derivative of 1 is 0, and the derivative of 10 is squared. Because we have a negative sign, we're going to write negative sesh squad of x in the numerator, right, because se squared is the derivative of change. And now considering our denominator once gained the derivative of 1 is 0, then we have a negative sign. Let's add it, and the derivative of the hyperbolic cotangent is negative cos squared, so we get negative of negative cos squared. Well done. So now, let's notice that we can cancel out our negative signs in the denominators. We can just write cosse squad of X. And we can factor out the negative signs, so we get negative limits. As x approaches infinity of Sash squared Of x divided by cosse squared of x. Now, let's recall that sash divided by cosse change, right? So we can also rewrite it as negative limit. As x approaches infinity of t. Squared of X Now let's remember once again that as x approaches infinity tinge of X approaches 1, right, so we're going to get negative or 1 squad, which is basically equal to -1. Therefore, our answer corresponds to the answer choice a. Thank you for watching.

88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.

lim x → ∞ (1 − coth x) / (1 − tanh x)

Key Concepts

Hyperbolic Functions (coth and tanh)

Limits at Infinity

l’Hôpital’s Rule

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