- #1
chemic_23
- 44
- 0
Homework Statement
what is the limit of
How to calculate the following limit - I'm stuck
In summary, the conversation discusses finding the limit of a function as x approaches 9. The attempted solution involves using the fact that a^2-b^2 can be written as (a+b)(a-b). By treating the numerator and denominator of the original function as a-b, the limit can be simplified to (1)/(2), which ultimately equals 4.
what is the limit of
- #2
sara_87
- 763
- 0
I would do it this way:
remember that:
a[tex]^{2}[/tex]-b[tex]^{2}[/tex]=(a+b)(a-b)
so
a-b=[tex]\frac{(a^2-b^2}{(a+b)}[/tex]
so take the numerator and denominator of your problem and treat each as a-b
so:
[tex]\sqrt{x}-3[/tex] = [tex]\frac{x-9}{\sqrt{x}+3}[/tex] (1)
and
[tex]\sqrt{1+\sqrt{x}}-2[/tex] = [tex]\frac{\sqrt{x} - 3}{\sqrt{1+\sqrt{x}}+2}[/tex] (2)
so your limit is now (1)/(2): which gives:
[tex]\frac{x-9}{\sqrt{x}+3}\times\frac{\sqrt{1+\sqrt{x}}+2}{\sqrt{x}-3}[/tex]
Is this clear?
Do you know how to continue from here?
- #3
chemic_23
- 44
- 0
continuation...
thus, the limit of
- #4
Dick
Science Advisor
Homework Helper
- 26,258
- 623
Correct.
FAQ: How to calculate the following limit - I'm stuck
How do I determine the limit of a function?
To calculate the limit of a function, you need to plug in values of x that are approaching the given limit from both the left and the right sides. If the function approaches the same value from both sides, then that value is the limit of the function.
What if the function is undefined at the given limit?
If the function is undefined at the given limit, you will need to factor or simplify the expression to see if you can cancel out the undefined term. If not, you can try using L'Hospital's rule or graphing the function to estimate the limit.
Can I use substitution to find the limit?
Substitution is a useful technique for evaluating limits, but it can only be used if the function is continuous at the given limit. If the function has a discontinuity at the limit, substitution will not provide an accurate answer.
What is the difference between a one-sided and two-sided limit?
A one-sided limit only considers values approaching the given limit from one direction (either the left or the right side), while a two-sided limit considers values approaching from both directions. The limit may be different depending on which direction you approach it from.
Are there any shortcuts or tricks for calculating limits?
There are no universal shortcuts for calculating limits, as it depends on the specific function and limit being evaluated. However, there are some common techniques such as factoring, simplifying, or using L'Hospital's rule that can help in certain cases. It is important to be familiar with these techniques and to practice using them to improve your limit-solving skills.