Solved Problems on Limits and Continuity

Overview of Problems 1 2 4 3 5 6 7 8 9 10

Overview of Problems 11 12 13 14 15

Main Methods of Limit Computations 1 The following undefined quantities cause problems: 2 In the evaluation of expressions, use the rules If the function, for which the limit needs to be computed, is defined by an algebraic expression, which takes a finite value at the limit point, then this finite value is the limit value. 3 If the function, for which the limit needs to be computed, cannot be evaluated at the limit point (i.e. the value is an undefined expression like in (1)), then find a rewriting of the function to a form which can be evaluated at the limit point. 4

Main Computation Methods Frequently needed rule 1 Cancel out common factors of rational functions. 2 If a square root appears in the expression, then multiply and divide by the conjugate of the square root expression. 3 Use the fact that 4

Continuity of Functions Functions defined by algebraic or elementary expressions involving polynomials, rational functions, trigonometric functions, exponential functions or their inverses are continuous at points where they take a finite well defined value. 1 A function f is continuous at a point x = a if 2 Used to show that equations have solutions. The following are not continuous x = 0: 3 4 Intermediate Value Theorem for Continuous Functions If f is continuous, f(a) < 0 and f(b) > 0, then there is a point c between a and b so that f(c) = 0.

Limits by Rewriting Problem 1 Solution

Limits by Rewriting Problem 2 Solution

Limits by Rewriting Problem 3 Solution Rewrite

Limits by Rewriting Problem 4 Solution Rewrite Next divide by x.

Limits by Rewriting Problem 5 Solution Rewrite Next divide by x.

Limits by Rewriting Problem 6 Solution

Limits by Rewriting Problem 7 Solution Rewrite:

Limits by Rewriting Problem 8 Solution Rewrite:

Limits by Rewriting Problem 9 Solution Rewrite

Limits by Rewriting Problem 9 Solution (cont’d) Rewrite Next divide by x. Here we used the fact that all sin(x)/x terms approach 1 as x  0.

Continuity Problem 11 Solution

Continuity Problem 12 Solution

Continuity Problem 13 Solution

Continuity Problem 14 Answer Removable Not removable

Continuity Problem 15 Solution By the intermediate Value Theorem, a continuous function takes any value between any two of its values. I.e. it suffices to show that the function f changes its sign infinitely often.