27. Use the definitions of the hyperbolic functions to find...

27. Use the definitions of the hyperbolic functions to find each of the following limits.\ a. $\lim_{x \to \infty} \tanh x$\ Answer\ b. $\lim_{x \to -\infty} \tanh x$\ Answer\ c. $\lim_{x \to \infty} \sinh x$\ Answer\ d. $\lim_{x \to -\infty} \sinh x$\ Answer\ e. $\lim_{x \to \infty} \sech x$\ Answer\ f. $\lim_{x \to \infty} \coth x$\ Answer\ g. $\lim_{x \to 0^+} \coth x$

Transcript

00:01 For part a, to find a limit as x approaches infinity of hyperbolic tangent of x, we can write this using its exponential form.

00:09 Then, if we multiply and divide by e to the negative x, this simplifies to the limit is x approaches infinity of 1 minus 1 over 8 to the 2x, all over 1 plus 1 over 8 to the 2x.

00:22 Now, if we substitute in infinity, then 1 over 8 to the 2x becomes a 0 for a final answer of 1.

00:31 If we're looking at the same limit, but now as x approaches negative infinity of hyperbolic tangent of x, once again, we'll rewrite hyperbolic tangent of x in its exponential form, and this time we'll multiply and divide by e to the x.

00:45 So we get the limit as x approaches negative infinity of e to the 2x minus 1 over e to the 2x plus 1.

00:52 Since x is approaching negative infinity, e to the 2x approach 0, so we get a final answer of negative 1.

01:00 Now, if we're looking at the limit as x approaches infinity of cinch of x, we can rewrite cinch of x in its exponential form, and now we can break this up with the two fractions.

01:10 So we're looking at the limit as x approaches infinity of e to the x over two minus 1 over 2, e to the x to the x goes to infinity, but 1 over e to the x goes to 0.

01:24 So the limit here is infinity.

01:27 Now if we're looking at the limit as x approaches negative infinity of cinch of x, once again, we can also break up cinch of x into two separate fractions, as we did in the previous case.

01:42 And now since we're approaching negative infinity, e to the x becomes a zero, and one over e to the x becomes an infinity.

01:51 So our limit now is negative infinity.

01:54 For part e, we're asked to find the limit as x approaches infinity of hyperbolic secan of x.

02:03 Using the exponential form and multiplying and dividing by e to the negative x, this becomes the limit as x approaches infinity of 2e to the negative x over 1 plus e to the negative 2x...

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