How can I solve a one-sided limit without using l'Hopital's rule?

In summary, a one-sided limit is a concept in calculus that describes the behavior of a function as the input approaches a specific value from either the left or the right side. To calculate a one-sided limit, the value being approached is substituted into the function and the resulting value is observed. This is different from a two-sided limit, which considers the behavior from both directions and can only exist if the one-sided limits are equal. One-sided limits can still exist even if the function is undefined at the limit value. They are important in calculus for analyzing function behavior, determining continuity, and defining derivatives and integrals.
  • #1
squirrelschaser
14
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Homework Statement

Find the ## lim _{x-> -1+} sqrt(x^2-3x)-2/|x+1| ##

Homework Equations

The Attempt at a Solution



I can only solve it using l'hopital rule and would like to know the steps of solving it without using it.

## lim _{x->-1+} (2x-3)/|1|= -5/4 ##
 
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  • #2
squirrelschaser said:

Homework Statement

Find the ## lim _{x-> -1+} sqrt(x^2-3x)-2/|x+1| ##

Homework Equations

The Attempt at a Solution



I can only solve it using l'hopital rule and would like to know the steps of solving it without using it.

## lim _{x->-1+} (2x-3)/|1|= -5/4 ##
Multiply the expression by 1 in the form of the conjugate of the numerator over itself. The |x + 1| factor in the denominator can be replaced by x + 1, since x is to the right of -1, so x + 1 > 0. If the limit had been as x approaches -1 from the left you have to replace |x + 1| by -(x + 1).
 
  • #3
squirrelschaser said:

Homework Statement

Find the ## lim _{x-> -1+} sqrt(x^2-3x)-2/|x+1| ##

Homework Equations

The Attempt at a Solution



I can only solve it using l'hopital rule and would like to know the steps of solving it without using it.

## lim _{x->-1+} (2x-3)/|1|= -5/4 ##

I suppose you mean$$
\lim_{x\to -1^+}\frac{\sqrt{x^2-3x}-2}{|x+1|}$$which is not what you wrote. Anyway since ##x>-1## you can write ##|x+1|=x+1##. Try rationalizing the numerator and see if you can get it then.

[Edit] Mark44 must type faster than I do.
 
Last edited:
  • #4
I'm dumb. Much thanks.
 
  • #5
No, you are careless- that, at least, is curable!
 

FAQ: How can I solve a one-sided limit without using l'Hopital's rule?

1. What is a one-sided limit?

A one-sided limit is a concept in calculus that refers to the behavior of a function as the input approaches a specific value from either the left or the right side. It is denoted by the use of a plus or minus sign, indicating whether the input is approaching from the positive or negative direction.

2. How is a one-sided limit calculated?

To calculate a one-sided limit, you must first substitute the value that the input is approaching into the function. Then, evaluate the function to see what value it approaches from that particular direction. This can be done algebraically or by creating a table of values and observing the trend.

3. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of a function as the input approaches a specific value from one direction, whereas a two-sided limit considers the behavior from both the left and right side. This means that a two-sided limit can only exist if the one-sided limits from both directions are equal.

4. Can a one-sided limit exist even if the function is undefined at the limit value?

Yes, a one-sided limit can exist even if the function is undefined at the limit value. This is because the limit is based on the behavior of the function as the input approaches the limit value, not necessarily the actual value of the function at that point.

5. Why are one-sided limits important in calculus?

One-sided limits are important in calculus because they allow us to analyze the behavior of a function at a specific point. They also help us determine if a function is continuous at a particular point, which is a key concept in calculus. Additionally, one-sided limits are used in the definition of derivatives and integrals, which are fundamental concepts in calculus.

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