Understanding Limits in Calculus - Prof. William Case, Study notes of Calculus

Download Understanding Limits in Calculus - Prof. William Case and more Study notes Calculus in PDF only on Docsity! Section 9.1 Limits Example of limits Find the limit: 1lim 2 1 + → x x Find the limit of the function as x approaches 1 x .9 .99 .999 1.1 1.01 1.001 )(xf 1.810 1.980 1.998 2.210 2.020 2.002 The values in the table seem to approach 2 as x approaches 1. There the limit of function as x approaches 1 is 2. This is written symbolically as 21lim 2 1 =+ → x x Definition 1: )()(lim cfxf cx = → Thus the limit in the above example can be found by using definition 1, 21111)1()(lim 2 1 =+=+== → fxf x Example 1 Find 3 3 lim x x→ Solution: 273lim 33 3 == → x x Example 2 Find xx x 83lim 2 4 + −→ Solution: 16324832)16(3)4(8)4(383lim 22 4 =−=−=−+−=+ −→ xx x Example 3 Find 1 3 lim 1 −→ xx Solution: Undefined xx 0 3 11 3 1 3 lim 1 = − = −→ Limit does not exist Example 4 Find 7lim 5 − → x x Solution: Undefinedx x 2757lim 5 −=−=− → Limit does not exist Example 5 Find 9 3 lim 23 − − → x x x Solution: 6 1 33 1 3 1 lim )3)(3( 3 lim 9 3 lim 3323 = + = + = +− − = − − →→→ xxx x x x xxx In example 5, if you substitute 3 in for x before you reduce the polynomial down, you will get an undetermined form. 0 0