The linear function ( x) = mx + b, for finite m 0, is a

The linear function ℓ(x) = mx + b, for finite m ≠ 0, is a slant asymptote of f(x) if

a. Use a graphing utility to make a sketch that shows ℓ(x) = x is a slant asymptote of f(x) = x tanh x. Does f have any other slant asymptotes?

b. Provide an intuitive argument showing that f(x) = x tanh x behaves like ℓ(x) = x as x gets large.

c. Prove that ℓ(x) = x is a slant asymptote of f by confirming

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lim ( f(x) – €(x)) = 0. lim (x tanh x – x) = 0.