Weekly Math Problem #72

Limits. I found this week's problem in my personal archives. 😁 I'm inspired by students I'll be working with that are learning limits. I can recall learning the concept of what a limit is and how to evaluate the limit of various types of functions. Honestly, I learned limits more after my Calculus 1 course as tutor than when I was enrolled in the course. Repetition really is the key 🗝 for me. Seriously, if I don't use it, I can definitely lose it. **Whispering** "This is why I started my blog." 😊

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     Substitution        
        ✔️     Simplifying rational expressions
        ✔️     Web Resource: Limits (Evaluating)

             

WMP! #72 want us to...

Happy solving!

Check back on Saturday, February 5th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

To begin the process of finding the limit, we'll use the substitution method. 

Unfortunately, the substitution method yields an indeterminate form expression. What does that mean? 🤷🏿‍♀️ This means that the method used doesn't tell us whether or not the limit exists. Another method for evaluating the limit needs to be used to determine whether or not the limit exists.

The numerator of the expression is a cubic polynomial that can be factored. The form is a difference of cubes. When the cubic expression is factored, you will see that one of the factors matches the expression in the denominator. **Yes!** So, we'll simplify the expression. This time, the substitution method will work.

Nice. The limit does exist, and it's equal to 3. 

Another method that can be used to evaluate a limit that yields and indeterminate form is L'Hôpital's Rule. It involves using derivatives and yields the same results. Take a look...

I didn't do it here, but you can graph the original expression and the simplified expression to see what their graphs look like. Then verify that as x approaches 1 from the left and right, the output value is 3.

▪️ Were you able to find the limit?
▪️ Did you do something else to find the limit? If so, please share. (No judgment.)

▪️ Let me know what you thought about this week's problem in the comments section. 

Thank you for solving with me this week. 😊
Up next...WMP! #73. 

Cheers!

The Younge Lady

Weekly Math Problem #50

Limits. For the most part, I understand limits. It reminds me of integration...where I get the idea but most of the work is knowing the techniques and when to implement them. Such is limits. There are different techniques and knowing when to implement them is important...well...it's important in math. If you're not into math, you most likely don't care. 🤷🏿‍♀️

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to find limits involving radical expressions 

Here is WMP! #50:

Happy solving!

Check back on Friday, May 21st for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

Just when you think you have it all figured out...you don't, sometimes. 😩 Let's get into this solution. 

The first thing I did was substitute -1 into the expression to see if a value pops out...but I was met with an indeterminate form.

 

The indeterminate form doesn't help me so, I tried another method--rationalizing the numerator--to see if that would help me with substitution.

With the rationalized form of the expression, I tried substitution again.

Lo and behold...indeterminate form again. 😩 Now, I need another method. This one, I know, won't fail me--L'Hôpital's Rule.

Ahhh...an answer, finally! All of that work to come up with 1/6. 😅🤷🏿‍♀️

▪️ How do you solve this problem??

▪️ Did you go straight to L'Hôpital's Rule??

▪️ Let me know what you thought about this week's problem in the comments section.

Thanks for solving with me this week!

Up next, WMP! #51. 👍🏿

Cheers!

The Younge Lady

Weekly Math Problem! #5

Improper Integrals. A calculus thing. There's just something about solving integrals that I find very intriguing and that's probably because it requires more thought and "tricks" to get the solution, as compared to taking derivatives. But that's also the annoying thing--sometimes trying to figure out what sort of "tricks" are needed to help you figure the integral out. 🙄 

Like verifying trigonometric identities (see WMP! #3), these type of problems are also like puzzles to me. To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Methods for solving integrals

    ✔️  Finding limits 

    ✔️  Using logarithmic rules 

It's been a super long while since I've solved an improper integral so...here goes WMP! #5... 

Happy solving! Check back on Friday, June 5th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

I don't know how much calculus you remember. I don't know how many limit rules you remember. I don't how much you remember about exponential functions. But, if you're like me, you didn't remember enough right off the top of your head to get this problem done quickly. 🤷🏿‍♀️ 🥴 

◾️ Was your solution this long?? (...Or is it just me??) 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

Coming up next...WMP! #6 👀

Cheers!

The Younge Lady

Weekly Math Problem! #1

It's SPRING! It's been spring for over a month now, so it's about time I get these problems going. (I know..I know...it's been a long wait. However, I have an excuse. It's the same as everybody else....and you know what it is. 👀) For a little more info on these problems, check out my previous post. As promised, The Younge Lady's weekly math problems are here! So, without further ado, here is WMP #1:

Limits Involving Trigonometric Functions. This week's question came from a tutoring session that happened a couple months ago and is typically learned in a Calculus I course. To solve it in completion, you need to recall the following math skills:

     ✔️  How to deal with fractions 
     ✔️  How to deal with trigonometric functions 
     ✔️  Limit properties
     ✔️  Finding derivatives of trigonometric functions 

I will say that there are two ways to do this problem. Ready?

Happy solving! Check back on Friday, May 8th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

As previously mentioned, there are two ways to find this limit. The first method is algebraic in nature; it is not necessary to know derivatives to find this limit. Limits are most often taught before derivatives are, so algebraic methods are explored. Once derivatives and other topics have been taught, limits are re-explored at some point and the method of L'Hôpital's rule is used to evaluate limits that yield indeterminate forms when evaluated at the specified value.

Algebraic Method

L'Hôpital's Rule Method

That's pretty much it. Two different methods, same answer. That's one of my favorite things about mathematics--one problem can be solved multiple ways. 

◾️ Did you arrive at the same answer I have above?? 

◾️ How did you find the limit?? 

◾️ When was the last time you did a limit problem??

◾️ Comment below with your responses and let me know what you thought about this week's problem.

Cheers!

The Younge Lady