Analysis: Limits, strictly increasing, differentiability

In summary, the conversation discusses two problems. The first problem is a graph where b is false and the limit of f(x) could be negative infinity or alternate between large positive and large negative values as x approaches infinity. The second problem involves a limit involving a graph, which the speaker believes looks okay. The speaker also mentions the possibility that their professor wants them to write out their steps more clearly.
  • #1
1,656
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I've worked all of these out. I'm mostly confident I did them correctly, but I'm prone to overlook subtleties or counterexamples sometimes.

http://i111.photobucket.com/albums/n149/camarolt4z28/1ab-1.png [Broken]

http://i111.photobucket.com/albums/n149/camarolt4z28/1gf2-1.png [Broken]
 
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  • #2
1. b is false.

Is f(x) positive ?

I'll look at the other problem soon.
 
  • #3
SammyS said:
1. b is false.

Is f(x) positive ?

I'll look at the other problem soon.

Oh. You're saying the limit could be negative infinity, too, not just positive infinity. That's true.
 
  • #4
Or, f(x) could alternate from large positive to large negative values as x → ∞.
 
  • #5
The second link looked OK to me.
 
  • #6
SammyS said:
The second link looked OK to me.

Good catch on (b). Thanks for the review!
 
  • #7
Its possible your professor wants you to write out your steps a bit more.
 
  • #8
deluks917 said:
Its possible your professor wants you to write out your steps a bit more.

For which ones?
 

1. What is the concept of limits in analysis?

Limits in analysis refer to the value that a function approaches as its input variable approaches a particular value. It is used to describe the behavior of a function near a specific point and is an essential tool in calculus and mathematical analysis.

2. How do you determine if a function is strictly increasing?

A function is strictly increasing if for any two points on the graph, the y-value of the second point is greater than the y-value of the first point. This means that as the input variable increases, the output variable also increases without any dips or plateaus. To determine if a function is strictly increasing, you can take the derivative of the function and check if it is always positive.

3. What is the significance of differentiability in analysis?

Differentiability is a key concept in analysis that describes the smoothness of a function. A function is differentiable at a point if it has a well-defined derivative at that point. This allows us to study the behavior of a function and its rates of change, which is essential in many real-world applications.

4. How do you find the derivative of a function?

The derivative of a function is a measure of its instantaneous rate of change. It can be found by taking the limit of the difference quotient as the distance between two points on the graph approaches zero. Alternatively, you can also use differentiation rules and techniques to find the derivative of a function.

5. Can a function be both continuous and not differentiable?

Yes, a function can be continuous at a point but not differentiable at that point. This occurs when the function is not smooth or has a sharp corner or cusp at that point. A classic example is the absolute value function, which is continuous but not differentiable at x=0.

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