103Lab06SolnUpdated.pdf - MA103 Lab Report 6 Shapes of...

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MA103 Lab Report 6 - Shapes of Curves Name: Student Number: Fall 2020 1. [6 marks ] Recall the class of functions called hyperbolic functions, one of which is tanh( x ) = e x - e - x e x + e - x . (a) Identify the indeterminate form of lim x →∞ tanh x . lim x →∞ tanh( x ) has the indeterminate form of . (b) Apply L’Hospital’s rule once to lim x →∞ tanh( x ) but do not evaluate the resulting limit. lim x →∞ tanh x = lim x →∞ e x - e - x e x + e - x H = lim x →∞ e x + e - x e x - e - x (c) Notice that the result from part (b) has another indeterminate form. Apply L’Hospital’s rule a second time but do not evaluate the resulting limit. lim x →∞ e x + e - x e x - e - x H = lim x →∞ e x - e - x e x + e - x = lim x →∞ tanh x (d) Looking at the result in (c), why would we not go ahead and apply L’Hospital’s Rule one more time? The result in (c) is the same as the initial function. L’Hospital’s rule will keep us going around in circles. (e) Re-write lim x →∞ tanh( x ) using the substitution t = e x and then evaluate the limit using L’Hospital’s rule. Note that when x → ∞ then t = e x → ∞ as well.

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