Question
Here is the question : GRADE 12 CALCULUS: WHAT IS THE LIMIT OF 1/(X – 1) AS X GOES TO INFINITY?
Option
Here is the option for the question :
- Infinity
- 1
- 0
- Negative infinity
The Answer:
And, the answer for the the question is :
Explanation:
Limits, which provide functions values when no values are given, are taught to students as they progress through the calculus curriculum.
In Grade 12 calculus, students delve into the concept of limits, which is a fundamental idea in calculus that allows us to understand the behavior of functions as their input values approach a particular value, such as infinity. Limits play a crucial role in various areas of mathematics and have wide-ranging applications in physics, engineering, and economics. Let’s explore an example where we consider the limit of (1/(x – 1)) as (x) approaches infinity. By applying the principles of limits and performing the necessary calculations, we find that the limit is equal to 0.
The limit of a function represents the value that the function approaches as the input values get arbitrarily close to a particular value. In this case, we are interested in finding the limit of (1/(x – 1)) as (x) approaches infinity. To determine the limit, we evaluate the behavior of the function as (x) becomes larger and larger.
As (x) approaches infinity, the denominator (x – 1) becomes increasingly larger. When the denominator grows without bound, the value of the entire fraction (1/(x – 1)) approaches 0. This is because any positive number divided by a very large positive number results in a value that approaches 0.
To formalize this, we can express the limit as:
(\lim_{{x \to \infty}} \frac{1}{{x – 1}} = 0).
This notation indicates that as (x) approaches infinity, the value of (1/(x – 1)) approaches 0.
Understanding limits is essential in calculus as they provide a precise way to describe and analyze the behavior of functions. Grade 12 students who develop a solid understanding of limits lay the foundation for more advanced topics such as derivatives, integrals, and differential equations.
Limits are encountered in various real-life scenarios, including physics, economics, and optimization problems. They are used to describe rates of change, determine maximum and minimum values, and analyze the behavior of systems. By mastering the concept of limits, students gain the ability to model and solve real-world problems that involve continuous change and approximation.
Furthermore, understanding limits enhances students’ mathematical and analytical skills. It enables them to analyze the behavior of functions, determine continuity, and evaluate the convergence or divergence of sequences and series. By studying limits, students also develop a deeper appreciation for the interconnectedness of different areas of mathematics and the power of calculus in describing and predicting real-world phenomena.
students will encounter more complex limit problems involving other types of limits such as limits at finite values, limits from the left and right, and limits involving trigonometric, exponential, and logarithmic functions. These advanced concepts further expand students’ mathematical toolkit and equip them with valuable analytical skills for a wide range of applications.
in Grade 12 calculus, understanding limits and evaluating them as input values approach specific values is a fundamental skill. By analyzing the behavior of functions as the input values become arbitrarily close to a particular value, we can determine the limit. In the case of (1/(x – 1)) as (x) approaches infinity, the limit is determined to be 0. Mastering the concept of limits not only improves mathematical proficiency but also enhances students’ problem-solving abilities and provides them with a powerful tool for analyzing and understanding the behavior of functions in various contexts.