Calculus, left hand, right hand limits.

In summary, When x approaches 3, the limit does not exist because the limits are different from the left than the right.
  • #1
charmedbeauty
271
0

Homework Statement



Its just a general query about problems along these lines...

f(x)=|x^2+3x-18|/(x-3) and a =3, discuss the limiting behaviour of f(x) as x→a^+, as x→a^- and as x→a.



Homework Equations





The Attempt at a Solution



So my basic solution to these types of problems are picking a number very close to 3 on either side of the number line, ie, 2.999 and 3.001 and then calculating to see what f9x) approaches given these values of x. The general answer I get is that x is negative on one side and positive on the other therefore as x→a the lim does not exist since lim x→a^- ≠ x→a^+. Although I have a test coming up and I know these types of questions are going to be involved, but I can't use a calculator in tthe test.

So my question is, do I need to state what f(x) approaces from both sides or is it sufficient to say its negative, then positive... therefore the lim does not exist.

The only reason I don't say what it approaches is because its hard to calculate ie what |(2.99999)^2 + 3(2.99999) -18|/ (2.99999 -3), is in my head.

so 1) do I need to state what it approaches... and 2) is there an easy way to calculate the squares and cubes etc of large decimal numbers is ie (2.99999999)^n?

Any help or thoughts greatly appreciated!
 
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  • #2
Try breaking up the numerator, as you would typically do for this type of problem.

The only problem now is that you have absolute values on the top. You can drop the absolute values to form two different cases where as x approaches 3, you keep the numerator positive. Then you can divide out the denominator and calculate the limit by plugging in x=3. Your intuition in right, the limit doesn't exist because the limits are different from the left than the right. By doing this method you can show exactly how they are different.
 
  • #3
charmedbeauty said:

Homework Statement



Its just a general query about problems along these lines...

f(x)=|x^2+3x-18|/(x-3) and a =3, discuss the limiting behaviour of f(x) as x→a^+, as x→a^- and as x→a.

Homework Equations



The Attempt at a Solution



So my basic solution to these types of problems are picking a number very close to 3 on either side of the number line, ie, 2.999 and 3.001 and then calculating to see what f9x) approaches given these values of x. The general answer I get is that x is negative on one side and positive on the other therefore as x→a the lim does not exist since lim x→a^- ≠ x→a^+. Although I have a test coming up and I know these types of questions are going to be involved, but I can't use a calculator in tthe test.

So my question is, do I need to state what f(x) approaces from both sides or is it sufficient to say its negative, then positive... therefore the lim does not exist.

The only reason I don't say what it approaches is because its hard to calculate ie what |(2.99999)^2 + 3(2.99999) -18|/ (2.99999 -3), is in my head.

so 1) do I need to state what it approaches... and 2) is there an easy way to calculate the squares and cubes etc of large decimal numbers is ie (2.99999999)^n?

Any help or thoughts greatly appreciated!
Without the calculator, it's difficult to tell the sign of [itex]x^2+3x-18[/itex] , near x = 3.

Try factoring [itex]x^2+3x-18[/itex]. You get [itex](x-3)(x+6)\,.[/itex]

It should be easy to determine the sign of [itex](x-3)(x+6)[/itex] when x is a little greater than 3 and when x is a little bit less than 3.
 
  • #4
Ok thanks I have worked out that you can cancel out terms in the numerator and denominator and then just plug in the approaching values...
Thanks for the input
 

1. What is a limit in calculus?

A limit in calculus is a fundamental concept that is used to describe the behavior of a function as it approaches a certain point. It is denoted by the symbol "lim" and is used to determine the value that a function approaches as its input variable gets closer to a specific value.

2. What is a left hand limit?

A left hand limit is the limit of a function as its input variable approaches a specific value from the left side. It is denoted by the symbol "lim" with a negative sign (-) above the input variable. This limit considers only the values of the function that are less than the specific value being approached.

3. What is a right hand limit?

A right hand limit is the limit of a function as its input variable approaches a specific value from the right side. It is denoted by the symbol "lim" with a positive sign (+) above the input variable. This limit considers only the values of the function that are greater than the specific value being approached.

4. How do you calculate left and right hand limits?

To calculate a left or right hand limit, you need to plug in values that are approaching the specific value from the respective side into the function and see what value the function is approaching. This can be done algebraically or graphically, using a table of values or a graph of the function.

5. Why are left and right hand limits important in calculus?

Left and right hand limits are important in calculus because they help us understand the behavior of a function around a specific point. They allow us to determine if a function is continuous at a certain point or if there are any discontinuities present. They also help in finding the derivatives and integrals of functions, which are essential in many real-world applications.

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