Page 212 - DMTH201_Basic Mathematics-1

Unit 7: Limits

Notes
Example: Use the Sandwich Theorem to prove that

Solution:
For any x 0, we have

Hence

Since

then the Sandwich Theorem implies

Example: Consider the function

Use the Sandwich Theorem to prove that

Solution:
Since we are considering the limit when x gets closer to 0, then we may assume that |x| 1. In
4
this case, we have x x 2x . Hence for any x, we have
2
2
2
1 f(x) 1 + 2x .
Since then the Sandwich Theorem implies

7.5 Infinite Limits
Some functions “take off” in the positive or negative direction (increase or decrease without
bound) near certain values for the independent variable. When this occurs, the function is said to
have an infinite limit; hence, you write Note also that the function

has a vertical asymptote at x = c if either of the above limits hold true.

In general, a fractional function will have an infinite limit if the limit of the denominator is zero

and the limit of the numerator is not zero. The sign of the infinite limit is determined by the sign
of the quotient of the numerator and the denominator at values close to the number that the
independent variable is approaching.

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