Solved Problems: Finite Limits
Summary:
This class offers solved problems on the calculation of finite limits, accompanied by a video tutorial. Students are guided to solve the problems and compare their results with the provided solutions. Key techniques such as rationalization and the use of graphs are highlighted, encouraging different approaches to reach the correct answer.
Learning Objectives
By the end of this class, the student will be able to:
- Calculate finite limits of polynomial and rational functions.
- Use rationalization techniques to simplify limit calculations.
- Compare different solving methods and recognize the existence of multiple paths to the correct solution.
These solved exercises will help you master the calculation of finite limits. The solutions are in the video.
To enhance the learning experience, I have created this video with solved exercises on limit calculation, showing all the particularities that can be inferred from them.
Try to solve these exercises on your own and then compare your results.
Next to each exercise, you will find a link that leads to the part of the YouTube video where all the calculations are performed up to the solution.
Sometimes you will have to rationalize, and other times you will have to rely on the definition of a limit to justify an answer. Sometimes, even a graph can save you a lot of work when deciding which line of reasoning to follow.
I. Calculate the following finite limits:
- \displaystyle \lim_{x\to -1} (x^3+2x^2 - 3x - 4) [SOLUTION]
- \displaystyle \lim_{x\to 0} \dfrac{3^x - 3^{-x}}{3^x + 3^{-x}} [SOLUTION]
- \displaystyle \lim_{x\to 2} \dfrac{x-1}{x^2-1} [SOLUTION]
- \displaystyle \lim_{x\to 2} \dfrac{x^2-4}{x^2-5x+6} [SOLUTION]
- \displaystyle \lim_{x\to -1} \dfrac{x^2+3x+2}{x^2+4x+3} [SOLUTION]
- \displaystyle \lim_{x\to 2} \dfrac{x-2}{\sqrt{x^2-4}} [SOLUTION]
- \displaystyle \lim_{x\to 2} \dfrac{\sqrt{x-2}}{x^2-4} [SOLUTION]
II. Find \displaystyle\lim_{h\to 0} \frac{f(x+h)-f(x)}{h} for the following functions:
- \displaystyle f(x) = \dfrac{1}{x-2} [SOLUTION]
- \displaystyle f(x) = \sqrt{x-4} [SOLUTION]
- \displaystyle f(x) = \dfrac{x}{x+1} [SOLUTION]
Have you managed to solve them all on your own?
How different are your solutions from mine?
Remember that the path to solving a problem is not necessarily unique.