How can I calculate left and right-sided limits?

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In summary: If you mean it is the integer between x and x+1, then it is the ceiling.In summary, The conversation discusses the calculation of left and right-sided limits for various expressions, including \frac{x}{a}[\frac{b}{x}], \frac{b}{x}[\frac{x}{a}], and \frac{x}{\sqrt{|sinx|}} at the point x=0. The participants also discuss using the definition of absolute value and the behavior of \sin x near x=0 to determine these limits. They also consider using the floor and ceiling functions, but there is some confusion about which one to use.
  • #1
Phizyk
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Hi,
How can I calculate left and right-sided limits?
[tex]\frac{x}{a}[\frac{b}{x}][/tex]
[tex]\frac{b}{x}[\frac{x}{a}][/tex]
[tex]\frac{x}{\sqrt{|sinx|}}[/tex]
in point x=0.
Thanks for help.
 
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  • #2


What have you done? think about how the definition of absolute value and how [tex] \sin x [/tex] behaves when [tex] x \approx 0 [/tex].
 
  • #3


Phizyk said:
Hi,
How can I calculate left and right-sided limits?
[tex]\frac{x}{a}[\frac{b}{x}][/tex]
For x not equal to 0, this is just b/a and so has b/a as both right and left sided limits.
Or did you mean (x/a)|b/x|? In that case, you take left and right limits by looking at:
If x> 0 then |b/x|= |b|/x so (x/a)(|b|/x)= |b|/a
If x< 0 then |b/x|= -|b|/x so (x/a)(|b|/x)= -|b|/a

[tex]\frac{b}{x}[\frac{x}{a}][/tex]
Same comments

[tex]\frac{x}{\sqrt{|sinx|}}[/tex]
in point x=0.
Thanks for help.
The last one should be easy. Since sin(-x)= -sin(x), |sin(-x)|= |sin(x)| and the only difference between x< 0 and x> 0 is in the numerator.
 
  • #4


[tex][\frac{b}{x}][/tex] it is entier function. I can not solve second case... It is harder than first. Can I do [tex](\frac{x}{a}-1)\frac{b}{x}\leq{[\frac{x}{a}]\frac{b}{x}}\leq{\frac{b}{a}}[/tex] and use [tex]|f(x)-g|\leq{\epsilon}[/tex] so [tex]g=\frac{b}{a}[/tex]?
 
  • #5


Phizyk said:
[tex][\frac{b}{x}][/tex] it is entier function.
I don't know what that means.

I can not solve second case... It is harder than first. Can I do [tex](\frac{x}{a}-1)\frac{b}{x}\leq{[\frac{x}{a}]\frac{b}{x}}\leq{\frac{b}{a}}[/tex] and use [tex]|f(x)-g|\leq{\epsilon}[/tex] so [tex]g=\frac{b}{a}[/tex]?
Where did the "-1" in [tex]\frac{x}{a}-1[/itex] come from?
 
  • #6


[tex][\frac{b}{x}][/tex] the floor and ceiling functions.
[tex]x-1\leq{[x]}\leq{x}[/tex]
 
  • #7


Chose one! Does it mean the floor or the ceiling. It can't be both! If you mean [x] is the integer between x-1 and x, then it is the floor.
 

1. What is a limit in calculus?

A limit is a fundamental concept in calculus that represents the value that a function approaches as its input approaches a specific value. It is used to describe the behavior of a function near a particular point.

2. How do I calculate a left-sided limit?

To calculate a left-sided limit, you need to evaluate the function as the input approaches the given value from the left side. This can be done by plugging in values that are slightly smaller than the given value and observing the resulting outputs. The limit will be the value that the outputs approach as the inputs get closer and closer to the given value from the left side.

3. How do I calculate a right-sided limit?

To calculate a right-sided limit, you need to evaluate the function as the input approaches the given value from the right side. This can be done by plugging in values that are slightly larger than the given value and observing the resulting outputs. The limit will be the value that the outputs approach as the inputs get closer and closer to the given value from the right side.

4. Can a limit exist at a discontinuity?

No, a limit cannot exist at a discontinuity. A discontinuity occurs when there is a break in the graph of a function. In this case, the function does not approach a single value as the input approaches the given value from either side, so the limit does not exist.

5. How do I use a graph to find left and right-sided limits?

To find left and right-sided limits using a graph, you can observe the behavior of the function as the input approaches the given value from the left and right sides. A limit exists if the function approaches the same value from both sides. If the function approaches different values from each side, the limit does not exist.

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