Solved Problems on Limits and Continuity

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Solved Problems on Limits and Continuity
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Overview of Problems
2
1
4
3
5
6
7
8
9
10
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Overview of Problems
11
12
13
14
15
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Main Methods of Limit Computations
The following undefined quantities cause
problems
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In the evaluation of expressions, use the rules
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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.
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5
Main Computation Methods
Cancel out common factors of rational functions.
2
6
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
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Used to show that equations have solutions.
The following are not continuous x 0
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Intermediate Value Theorem for Continuous
Functions
If f is continuous, f(a) lt 0 and f(b) gt 0, then
there is a point c between a and b so that
f(c) 0.
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Limits by Rewriting
Problem 1
Solution
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Limits by Rewriting
Problem 2
Solution
9
Limits by Rewriting
Problem 3
Solution
Rewrite
10
Limits by Rewriting
Problem 4
Solution
Rewrite
Next divide by x.
11
Limits by Rewriting
Problem 5
Solution
Rewrite
Next divide by x.
12
Limits by Rewriting
Problem 6
Solution
13
Limits by Rewriting
Problem 7
Rewrite
Solution
14
Limits by Rewriting
Problem 8
Rewrite
Solution
15
Limits by Rewriting
Problem 9
Solution
Rewrite
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Limits by Rewriting
Problem 9
Solution(cont