Math 210 Infinite Limits and Continuity Homework

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Math 210: Written Homework 1, due Monday February 11th Name:________________________ Instructions: Write all explanations neatly. Use proper mathematical notation. Part of your grade will be based on quality of writing. 1) (30 pts) For parts a – d, evaluate the limit. a. (8) Compute this limit. You may use either method shown in class. lim⁡⁡ ⁡ −3𝑥 2 −5𝑥 𝑥→−∞ 𝑥 2 +7𝑥+10 1 b. (6) lim ( 𝑥 + 𝑒 𝑥 ) 𝑥→−∞ 2𝑥 2 + 5,⁡⁡⁡𝑥 < 0 c. (8) ⁡⁡⁡𝑓(𝑥) = { 3−5𝑥3 ,⁡⁡⁡𝑥 ≥ 0 1+4𝑥+𝑥3 i. lim 𝑓(𝑥) 𝑥→∞ ii. lim 𝑓(𝑥) 𝑥→−∞ d. (8) Use the Squeeze Theorem to compute the limit. SHOW ALL YOUR STEPS. (cos 𝑥)+2 ⁡⁡ 𝑥3 lim ⁡ 𝑥→∞ 2) (10 pts) Given the function, find the value of 𝑘 that makes the function continuous everywhere. 𝑓(𝑥) = { 7𝑥 − 2⁡,⁡⁡⁡⁡⁡𝑥 ≤ 1 𝑘𝑥 2 , 𝑥>1 3) (10 pts) On one graph, sketch the function with these given properties. You do not need to find a formula for the function. If there is a vertical asymptote, state its equation. If there is a horizontal asymptote, state its equation. 1 𝑓(1) = 0⁡⁡⁡ 𝑓(0) = − 2 lim − 𝑓(𝑥) = −∞⁡⁡⁡⁡⁡⁡ lim + 𝑓(𝑥) = ∞⁡⁡⁡⁡ 𝑥→−2 𝑥→−2 𝑓(4) = 3 ⁡⁡⁡⁡ lim 𝑓(𝑥) = 1⁡⁡⁡⁡ 𝑥→−∞
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