# Limits at infinity

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Find each of the following limits, with justification. If the limit does not exist, explain why. If there is an infinite limit, then explain whether it is ∞ or −∞. (a) limx→∞ (2√ x + x) 2 5 + x √ x (b) limx→∞ e x + 2x 5 x + 3 (c) lim x→−∞ x −1 + x −4 x−2 − x−3 (d) limx→∞ sin(x)e −x 2 (e) limx→∞ sin ln 1 x . 2. Let f(x) = e x + e −x e x − e−x . (a) Find the equation of all vertical asymptotes or explain why none exist. As justification for each asymptote x = a, calculate both the one-sided limits limx→a+ f(x) and limx→a− f(x). (b) Find the equation of all horizontal asymptotes or explain why none exist. 3. Section 2.6: 8, 10 4. Let f(x) = 3 x − 2 ln x . Evaluate the following: (a) limx→∞ f(x) (b) lim x→0+ f(x) 1 (c) lim x→1− f(x) (d) lim x→1+ f(x). 5. Let f(x) = 3 x 2 . (a) Calculate f 0 (a) using the limit definition of the derivative. (b) Find the equation of the tangent line to y = f(x) at the point (2, 3/4). 6. Let f(x) = 7 3 + e x . (a) Find the domain and the range of f(x). (b) Find the equation of all vertical asymptotes or explain why none exist. As justification for each asymptote x = a, calculate both the one-sided limits limx→a+ f(x) and limx→a− f(x). (c) Find the equation of all horizontal asymptotes or explain why none exist. (d) Find the inverse function f −1 (x). You don’t need to show that f is one-to-one.

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