EXAM 1 (T/F)/(Short Answer)

What are the 3 ways in which a limit may fail to exist?

-The left and right hand limits of f(x) as x approaches c are not equal -f(x) approaches infinity as x approaches c -f(x) oscillates without approaching any specific value as x approaches c

Describe 3 situations where the limit of f(x) as x approaches c does not exist.

-the function approaches different f(c) values from both sides of x=c -the function moves towards infinity as x approaches c (x=c is an asymptote) -the function oscillates without approaching a specific value and x approaches c

When x is near 0, (sinx)/x is near what value?

1

In your own words, what is a difference quotient?

A difference quotient is a kind of function which is designed to provide the average rate of change over a given period of time.

In your own words, what does it mean to "find the limit of f(x) as x approaches 3"?

Determine which value f(x) (the output) approaches as x (the input) approaches 3.

T/F: If the limit of f(x) as x approaches 5 is equal to infinity, then we are implicitly stating that the limit exists.

False, as a limit does not exist if f(x) approaches infinity as x approaches 5.

T/F: If the limit of f(x) as x approaches 1 from values less than 1 is equal to negative infinity, then the limit of f(x) as x approaches 1 from values greater than 1 is equal to infinity.

False, as it is not known how the function behaves as x approaches 1 from values greater than 1.

T/F: If f is defined on an open interval containing c, and the limit of f(x) as x approaches c exists, then f is continuous at c.

False, as it must be known whether f(c) is equal to the limit of f(x) as x approaches c before it can be stated that f is continuous at c.

T/F: If the limit of f(x) as x approaches 1 from values less than 1 is equal to 5, then the limit of f(x) as x approaches 1 from values greater than 1 is also equal to 5.

False, as the behavior of the function as x approaches 1 from values greater than 1 is not known, so the limit can not be assumed to be 5.

T/F: If the limit of f(x) as x approaches 1 from values less than 1 is equal to 5, then the limit of f(x) as x approaches 1 is also equal to 5.

False, as the behavior of the function as x approaches 1 from values greater than 1 is not known, so the limit can not be assumed to be 5.

T/F: If f is continuous on [a,b], then the limit of f(x) as x approaches a from values less than a is equal to f(a).

False, as values less than a are not included in the interval of continuity, so it cannot be known whether or not the limit of f as x approaches a from values less than a is equal to f(a).

T/F: The limit of f(x) as x approaches 5 is f(5).

False. It is not known whether x=5 is defined on the domain of f, nor is the behavior of the function around x=5 known.

In your own words, describe what the intermediate value theorem states.

For a continuous function with the values (a,f(a)) and (b,f(b)), where a<b and f(a)<(b), there lies some value (c,f(c)) at which a<c<b and f(a)<f(c)<f(b).

An expression in the form "0/0" is called:

Indeterminate

What functions have a constant rate of chanege?

Linear functions

Given a function y=f(x), in your own words describe how to find the units of f'(x).

Since a first order derivative describes the rate of change of a function, the units would be "y units/unit x". (For every unit x, y changes __ units)

Explain in your own words why the limit of b as x approaches c is equal to b.

Since b is a constant, no matter which value c is approached, f(c)=b. (b is just a horizontal line)

Explain in your own words why the limit of x as x approaches c is equal to c.

Since y=x is a 1:1 function, any input will produce an identical output. Therefore, as x approaches some value c, f(x) will approach the same value c.

You are given the following information: -the limit of f(x) as x approaches 1 is equal to 0 -the limit of g(x) as x approaches 1 is equal to 0 -the limit of (f(x)/g(x) as x approaches 1 is equal to 2 What can be said about the relative sizes of f(x) and g(x) as x approaches 1?

That f(x) is twice as large as g(x)

In your own words, describe what it means for a function to be continuous.

There are no points of discontinuity within the domain of the function.

What does it mean when we say that a certain function's behavior is 'nice' in terms of limits? What in particular is 'nice'?

To say a function's behavior is 'nice' in terms of limits is to say that f(c) is equal to the limit of f as x approaches c. (The limit and output of f as x approaches/equals c are identical)

T/F: If f is continuous at c, then the limit of f(x) as x approaches c exists.

True, as for f to be continuous at c, f must approach the same values from either side of c.

T/F: If the limit of f(x) as x approaches 1 is equal to 5, then the limit of f(x) as x approaches 1 from values less than 1 is also equal to 5.

True, as for the limit of f(x) to be equal to 5, both of the one-sided limits must be equal to 5.

T/F: The limit of (lnx) as x approaches 1 is 0. Use a theorem to defend your answer.

True, as ln(1) is equal to 0 (function behaves nicely). This is true because in Theorum 1.3.3, it states "the limit of lnx as x approaches c is equal to lnc.

T/F: If f is continuous at c, then the limit as f(x) approaches x approaches c from values greater than c is equal to f(c).

True, as since c is a point of continuity, f(c) must be equal to the limit as f(x) as x approaches c, which includes values greater than c.

T/F: The sum of continuous functions is also continuous

True, as the continuity of the sum of 2 continuous functions is a property of continuous functions.

T/F: The definition of the derivative of a function at a point involves taking a limit.

True, as the definition of a derivative is the limit as h approaches 0 of the difference quotient of the function.

T/F: If the limit of f(x) as x approaches 5 is equal to infinity, then f has a vertical asymptote at x=5.

True, as the definition of a vertical asymptote is that f(x) approaches infinity as x approaches c.

T/F: If the limit of f(x) as x approaches infinity is equal to 5, then we are implicitly stating the limit exists.

True, as the limit of a function as x approaches infinity is capable of existing.

Imagine a graph which visually demonstrates the squeeze theorum.

Two lines f(x) and h(x) "squeeze" g(x) at a point at which all 3 functions produce an identical output.

Let the limit of f(x) as x approaches 7 be equal to infinity. Explain how we know that f is/is not continuous at x=7.

We know that f is not continuous at x=7 because, even if f is defined at x=7, the limit as x=7 does not exist. Therefore, x=7 would be a point of discontinuity due to failing to meet the criteria for being continuous.