The Limit of a Function - Study Note

SECTION 2.2 THE LIMIT OF A FUNCTION each quantity, if it exists. If it does not exist, explain why. (a) lim� t�t� (b) lim� t�t� (c) lim t�t� (b) lim� f �x� � �� xl4 x l�3 4. For the function f whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. (a) lim f �x� (b) lim� f �x� (c) lim� f �x� xl0 xl3 x l3 (d) lim f �x� tl0 tl0 tl0 (d) lim� t�t� (e) lim� t�t� (g) t�2� (h) lim t�t� tl2 (f) lim t�t� tl2 tl2 tl4 (e) f �3� xl3 97 7. For the function t whose graph is given, state the value of 3. Explain the meaning of each of the following. (a) lim f �x� � � |||| y y 4 4 2 2 2 0 2 4 t 4 x 5. Use the given graph of f to state the value of each quantity, if it exists. If it does not exist, explain why. (a) lim� f �x� (b) lim� f �x� (c) lim f �x� xl1 xl1 (d) lim f �x� 8. For the function R whose graph is shown, state the following. xl1 (b) lim R�x� (c) lim � R�x� (d) lim � R�x� xl5 x l2 (e) f �5� xl5 (a) lim R�x� x l �3 x l �3 (e) The equations of the vertical asymptotes. y y 4 2 0 2 4 0 _3 x 2 x 5 6. For the function h whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. (a) lim � h�x� (b) lim � h�x� (c) lim h�x� x l �3 9. For the function f whose graph is shown, state the following. x l �3 x l �3 (d) h��3� (e) lim� h�x� (f) lim� h�x� (a) lim f �x� (b) lim f �x� (g) lim h�x� (h) h�0� (i) lim h�x� (d) lim� f �x� (e) lim� f �x� xl0 xl0 ( j) h�2� x l�7 x l0 xl6 xl2 (k) lim� h�x� (c) lim f �x� xl0 xl6 (f) The equations of the vertical asymptotes. (l) lim� h�x� x l5 x l�3 x l5 y y _7 _4 _2 0 2 4 6 _3 0 6 x x 10. A patient receives a 150-mg injection of a drug every 4 hours. The graph shows the amount f �t� of the drug in the blood- 98 |||| CHAPTER 2 LIMITS AND DERIVATIVES x 2 � 2x , xl �1 x � x � 2 x � 0, �0.5, �0.9, �0.95, �0.99, �0.999, �2, �1.5, �1.1, �1.01, �1.001 stream after t hours. Find lim f �t� 18. lim lim f �t� and tl 12� tl 12� and explain the significance of these one-sided limits. 19. lim xl0 f(t) 2 ex � 1 � x , x2 20. lim� x ln�x � x 2 �, xl0 300 x � �1, �0.5, �0.1, �0.05, �0.01 x � 1, 0.5, 0.1, 0.05, 0.01, 0.005, 0.001 21–24 Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. 150 0 4 8 12 16 t 21. lim sx � 4 � 2 x 22. lim tan 3x tan 5x 23. lim x6 � 1 x10 � 1 24. lim 9x � 5x x xl0 xl1 1�x ; 11. Use the graph of the function f �x� � 1��1 � e � to state the value of each limit, if it exists. If it does not exist, explain why. (a) lim� f �x� xl0 (b) lim� f �x� xl0 25–32 Determine the infinite limit. (c) lim f �x� xl0 xl0 xl0 25. lim x l�3� 12. Sketch the graph of the following function and use it to deter- mine the values of a for which lim x l a f �x� exists: 27. lim x l1 � 2�x if x � �1 if �1 � x � 1 f �x� � x �x � 1�2 if x � 1 x�2 x�3 26. 2�x �x � 1�2 28. lim� x l5 29. lim� ln�x 2 � 9� ex �x � 5�3 x l� lim� x csc x 32. lim� x l 2� 13–16 Sketch the graph of an example of a function f that x�2 x�3 30. lim� cot x x l3 31. lim x l�3� xl2 x 2 � 2x x � 4x � 4 2 satisfies all of the given conditions. 13. lim� f �x� � 2, xl1 14. lim� f �x� � 1, xl0 lim f �x� � 1, x l 2� 15. lim� f �x� � 4, xl3 f �3� � 3, f �1� � 1, lim f �x� � �1, lim f �x� � 0, x l 2� f �0� is undefined lim f �x� � 2, x l 3� lim f �x� � 3, x l 4� 1 1 and lim� 3 x l1 x � 1 x3 � 1 (a) by evaluating f �x� � 1��x 3 � 1� for values of x that approach 1 from the left and from the right, (b) by reasoning as in Example 9, and (c) from a graph of f . 33. Determine lim� x l1 x l 0� f �2� � 1, f �1� � 2 ; 34. (a) Find the vertical asymptotes of the function lim f �x� � 2, x l �2 f ��2� � 1 16. lim f �x� � 3, xl1 lim f �x� � �2, x l 1� y� lim f �x� � �3, x l 4� ; f �4� � �1 17–20 Guess the value of the limit (if it exists) by evaluating the x2 � 1 3x � 2x 2 (b) Confirm your answer to part (a) by graphing the function. 35. (a) Estimate the value of the limit lim x l 0 �1 � x�1�x to five ; decimal places. Does this number look familiar? (b) Illustrate part (a) by graphing the function y � �1 � x�1�x. function at the given numbers (correct to six decimal places). ; 36. (a) By graphing the function f �x� � �tan 4x��x and zooming x 2 � 2x 17. lim 2 , x � 2.5, 2.1, 2.05, 2.01, 2.005, 2.001, x l2 x � x � 2 1.9, 1.95, 1.99, 1.995, 1.999 in toward the point where the graph crosses the y-axis, estimate the value of lim x l 0 f �x�. (b) Check your answer in part (a) by evaluating f �x� for values of x that approach 0.