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Sample Questions for Exam 1 (Limits – Sections 2.1 to 2.5) 1. Sketch in the grid provided below the graph of a function f that satisfies all of the following conditions: lim f ( x) , lim f ( x) 0 , lim f ( x) 1 , lim f ( x) , x x 3 2. x lim f ( x) , x 3 x 3 For the following functions, locate any vertical asymptotes. Justify your answers using limits. a) f ( ) tan 1 b) g ( x) c) h( x) 1 ln(5 x) x2 x 1 x 3 2 3. Evaluate the following limits: a) lim x 6 b) x 6 x 6 x 2 2 x 8 lim 4 x 16 x2 c) t 2 4 lim 3 t 8 t 2 Difference of Cubes Identity: a 3 b3 a b a 2 ab b 2 d) lim x e) 3 h ln(sin x) lim x g) x 2 1 2 x 3 3 h 1 1 lim h 0 f) 3x5 2 x3 1 sin 2 x lim 2 x x 4. Show that the limit given below does not exist. 5. Let f be the function defined everywhere (i.e. for all real numbers) by f ( x) x x , where is the floor function defined as follows for any real number x : x = the greatest integer that is less than or equal to x . Investigate the limit of f as x approaches 1. Does the limit exist? If so, justify your answer.