limit and contuinity
Limit at infinity
The function has the limit as increases without bound (or, asapproaches infinity), written if can be made arbitrarily close to by taking large enough. Similarly, the function has the limit as decreases without bound (or, asapproaches negative infinity), written if can be made arbitrarily close to by taking to be negative and sufficiently large in absolute value.
One-sided limits
The function has the right-hand limit as approaches from the right written if the value of can be made as close to as we please by taking sufficiently close to (but not equal to) and to the right of. Similarly, the function has the left-hand limit as approaches from the left written if the value of can be made as close to as we please by taking sufficiently close to (but not equal to) and to the left of.
Example:
According to Weiss’s law of excitation of tissue, the strength of the electric current is related to the time the current takes to excite tissue by the formula where with and are positive constants.
a. Evaluate and interpret your result.
b. Evaluate and interpret your result.
(Note: The limit in part (b) is called the threshold strength of the current. Why?)
Solution
a.
As the time taken to excite tissue smaller and smaller, the strength of electric current gets stronger and stronger
b.
As the time taken to excite tissue smaller and smaller, the strength of electric current gets stronger and stronger
The intermediate value theorem
If is a continuous function on a close interval and is any number between and, then there is at least one number in such that.
Example:
Khairul is looking straight out a window of an apartment building at a high of 32 ft. from the ground. A boy throws a tennis ball straight up by the side of the building where the window is located. Suppose the height of the ball (measured in feet) from the ground at time is .
a. Show that and
b. Use the intermediate
The function has the limit as increases without bound (or, asapproaches infinity), written if can be made arbitrarily close to by taking large enough. Similarly, the function has the limit as decreases without bound (or, asapproaches negative infinity), written if can be made arbitrarily close to by taking to be negative and sufficiently large in absolute value.
One-sided limits
The function has the right-hand limit as approaches from the right written if the value of can be made as close to as we please by taking sufficiently close to (but not equal to) and to the right of. Similarly, the function has the left-hand limit as approaches from the left written if the value of can be made as close to as we please by taking sufficiently close to (but not equal to) and to the left of.
Example:
According to Weiss’s law of excitation of tissue, the strength of the electric current is related to the time the current takes to excite tissue by the formula where with and are positive constants.
a. Evaluate and interpret your result.
b. Evaluate and interpret your result.
(Note: The limit in part (b) is called the threshold strength of the current. Why?)
Solution
a.
As the time taken to excite tissue smaller and smaller, the strength of electric current gets stronger and stronger
b.
As the time taken to excite tissue smaller and smaller, the strength of electric current gets stronger and stronger
The intermediate value theorem
If is a continuous function on a close interval and is any number between and, then there is at least one number in such that.
Example:
Khairul is looking straight out a window of an apartment building at a high of 32 ft. from the ground. A boy throws a tennis ball straight up by the side of the building where the window is located. Suppose the height of the ball (measured in feet) from the ground at time is .
a. Show that and
b. Use the intermediate