# I have troubles finding the limit of this piecewise function

• El foolish Phenomeno
El foolish Phenomeno
Homework Statement
find the domain of the piece wise function , its limit at 2, and negative infinity , positive infinity and 1/2 and find a so that the function is continuous at 2
Relevant Equations
lim f(x) when x tends to y = L
I have troubles finding the limits at the designated points , should i only find the limit at infinity where f(x) has belongs to an interval containing inifinity? (sorry for english)
and for the a this is what i attempted. i am unsure.

Our textbook never talks about piecewise functions and their rules. If you havd any textbook recommandations or websites videos that can help me learn more abkut the calculus of piecewise functions. i will gladly take them

Thank you for your help and your time. It's greatly appreciated

El foolish Phenomeno said:
Our textbook never talks about piecewise functions and their rules. If you havd any textbook recommandations or websites videos that can help me learn more abkut the calculus of piecewise functions. i will gladly take them

There's no such thing as a "piecewise" function. It's just a regular function that is given by a different formula for different values of ##x##. There are, therefore, no different rules.

The key point is that the limits, continuity and differentiability of these functions may not be obvious. In general, you have to check explicitly at the points where the function's formula changes. Have you studied one-sided limits? Like ##\lim_{x \to 2^-}## and ##\lim_{x \to 2^+}##.

For the limits at ##\pm \infty## the usual definition applies. It should be clear that as ##x \to \infty## only one formula is relevant.

PeroK said:

There's no such thing as a "piecewise" function. It's just a regular function that is given by a different formula for different values of ##x##. There are, therefore, no different rules.

The key point is that the limits, continuity and differentiability of these functions may not be obvious. In general, you have to check explicitly at the points where the function's formula changes. Have you studied one-sided limits? Like ##\lim_{x \to 2^-}## and ##\lim_{x \to 2^+}##.

For the limits at ##\pm \infty## the usual definition applies. It should be clear that as ##x \to \infty## only one formula is relevant.
We have studied one-sided limits.

I don't know if i am getting you but from what i understand of your explanation "It should be clear that ##x \to \infty## only one formula is relevant." you meant that if x tends to +infinity i just need to study the formula defined in the interval [2, +infinity) , for negative infinity i should only use the the interval (-infinity, 1/2]. does the same logic apply to limits as tends to 2 and 1/2?

El foolish Phenomeno said:
We have studied one-sided limits.

I don't know if i am getting you but from what i understand of your explanation "It should be clear that t ##x \to \infty## only one formula is relevant." you meant that if x tends to +infinity i just need to study the formula defined in the interval [2, +infinity) , for negative infinity i should only use the the interval (-infinity, 1/2].
Yes to both questions.

El foolish Phenomeno said:
does the same logic apply to limits as tends to 2 and 1/2?
Not exactly. Since 1/2 and 2 are endpoints of intervals in which different formulas apply to get the two-sided limits at each of these points you'll need to determine the left- and right-side limits at each of these points, meaning that you will need to use the appropriate function formula for a left-side limit or a right-side limit.

BTW, your posted pictures are unreadable, at least by me. Your handwriting is very difficult to read, and the extremely poor lighting in the images make things worse. For reasons of legibility we discourage the use of images that show the work done.

El foolish Phenomeno said:
We have studied one-sided limits.
Good. That's what you need if you have a different formula either side of a point. A limit exists iff the one-sided limits exist and are equal. That's a theorem!
El foolish Phenomeno said:
I don't know if i am getting you but from what i understand of your explanation "It should be clear that ##x \to \infty## only one formula is relevant." you meant that if x tends to +infinity i just need to study the formula defined in the interval [2, +infinity) , for negative infinity i should only use the the interval (-infinity, 1/2]. does the same logic apply to limits as tends to 2 and 1/2?
Yes. You can prove these things if you know how. In general, if we are looking at the limit as ##x \to x_0##, we can assume that ##|x - x_0| < 1##, for example. That's sometimes useful.

Another example is where ##x_0 > 0##, we can assume that ##x > 0## when looking at the limit. If you are using the epsilon-delta definition, then the justification is that we can take ##\delta < x_0##. So that ##|x - x_0| < \delta \ \Rightarrow \ x > 0##. Does that make sense?

PeroK said:
Good. That's what you need if you have a different formula either side of a point. A limit exists iff the one-sided limits exist and are equal. That's a theorem!

Yes. You can prove these things if you know how. In general, if we are looking at the limit as ##x \to x_0##, we can assume that ##|x - x_0| < 1##, for example. That's sometimes useful.

Another example is where ##x_0 > 0##, we can assume that ##x > 0## when looking at the limit. If you are using the epsilon-delta definition, then the justification is that we can take ##\delta < x_0##. So that ##|x - x_0| < \delta \ \Rightarrow \ x > 0##. Does that make sense?
it's make sense , thank you

Mark44 said:
Yes to both questions.

Not exactly. Since 1/2 and 2 are endpoints of intervals in which different formulas apply to get the two-sided limits at each of these points you'll need to determine the left- and right-side limits at each of these points, meaning that you will need to use the appropriate function formula for a left-side limit or a right-side limit.

BTW, your posted pictures are unreadable, at least by me. Your handwriting is very difficult to read, and the extremely poor lighting in the images make things worse. For reasons of legibility we discourage the use of images that show the work done.
next time i will try to write everything in latex. Thanks for the help

PeroK said:
There's no such thing as a "piecewise" function. It's just a regular function that is given by a different formula for different values of ##x##. There are, therefore, no different rules.

A function which has different rules ("formulas") for different input values ("different values of ##x##") is what is called a "piecewise function". In other words, a piecewise function is one which is defined by different rules for different pieces of the real number line. The term "piecewise function" is common in pre-calculus algebra (at least within the US, wherein lies most of my experience).

Mark44
e_jane said:
A function which has different rules ("formulas") for different input values ("different values of ##x##") is what is called a "piecewise function". In other words, a piecewise function is one which is defined by different rules for different pieces of the real number line. The term "piecewise function" is common in pre-calculus algebra (at least within the US, wherein lies most of my experience).
My point is that is a vacuous definition that has no mathematical content. A function is a function. It should be a "function defined piecewise". Hence the OP's confusion that they are different from regular functions, with different properties.

PeroK said:
A function is a function. It should be a "function defined piecewise". Hence the OP's confusion that they are different from regular functions, with different properties.
Good point. I wish textbooks and (more) instructors took greater care on this topic, precisely to avoid that confusion.

PeroK

## 1. How do I find the limit of a piecewise function at a point where the function definition changes?

To find the limit of a piecewise function at a point where the definition changes, you need to evaluate the left-hand limit and the right-hand limit separately. If both limits exist and are equal, then the limit of the function at that point exists and is equal to that common value. If the limits are different, the limit does not exist at that point.

## 2. What should I do if the piecewise function has different expressions on either side of the point of interest?

If the piecewise function has different expressions on either side of the point of interest, you should calculate the limit from the left using the expression that applies for values less than the point, and the limit from the right using the expression that applies for values greater than the point. Compare these two limits to determine if the overall limit exists.

## 3. How do I handle piecewise functions with undefined points or discontinuities?

When dealing with piecewise functions that have undefined points or discontinuities, focus on the behavior of the function as it approaches the point of interest from both sides. Even if the function is undefined at the point itself, the limit can still exist if the left-hand and right-hand limits are equal.

## 4. Can I use direct substitution to find the limit of a piecewise function?

Direct substitution can be used to find the limit of a piecewise function only if the function is continuous at the point of interest. For points where the function definition changes, you need to evaluate the left-hand and right-hand limits separately instead of relying on direct substitution.

## 5. What if the piecewise function involves more complex expressions like polynomials or trigonometric functions?

If the piecewise function involves more complex expressions like polynomials or trigonometric functions, you should still evaluate the left-hand and right-hand limits separately. Use appropriate limit laws and techniques for each piece of the function, such as factoring, rationalizing, or applying trigonometric identities, to find these limits.

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