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LIMIT THEOREMS To obtain results in calculus, we will frequently operate
with limits. The proofs of theorems shown in this section will be omitted in
the interest of brevity. The theorems will be stated and examples will be
given. Assume that we have three simple functions of x. Further,
let these functions [ f (x), g(x), and h (x)] have separate
limits such that
Theorem 1. The limit of the sum of two functions is equal to the sum of the
limits:
This theorem may be extended to include any number of
functions, such as
EXAMPLE: Find the limit of
SOLUTION:
Theorem 2. The limit of a constant, c,
times a function, f(x), is equal to the constant, c, times the limit of the function:
EXAMPLE: Find the limit of
SOLUTION
Theorem 3. The limit of the product of two
functions is equal to the product of their limits:
EXAMPLE: Find the limit of
SOLUTION.
Theorem 4. The limit of the quotient of two functions is
equal to the quotient of their limits, provided the limit of the divisor is not
equal to zero:
EXAMPLE: Find the limit of
SOLUTION
PRACTICE PROBLEMS Find the limits of the following, using the theorem
indicated:
ANSWERS:
