In this article, we will explore the limit of the function ( f(x) = frac{3 sin(5x)}{sin(2x)} ) as ( x ) gets closer to 0. We’ll look at how the function behaves by examining its values near zero and using a graph to visualize it.
To grasp the limit, we first check the values of ( f(x) ) as ( x ) approaches 0 from both the negative and positive sides.
When ( x ) is slightly less than 0, we notice the following:
As ( x ) gets closer to 0 from the negative side, the function values increase and seem to approach 7.5.
Now, let’s see what happens as ( x ) approaches 0 from the positive side:
Similar to the negative side, as ( x ) approaches 0 from the positive side, the function values also converge towards 7.5.
From our observations, whether approaching from the left (negative side) or the right (positive side), the limit of ( f(x) ) as ( x ) approaches 0 seems to be 7.5. This consistency allows us to confidently state:
[lim_{x to 0} f(x) = 7.5]
To further support our findings, we can graph the function ( f(x) = frac{3 sin(5x)}{sin(2x)} ). By setting a suitable range for ( x ) around 0 and adjusting the ( y )-axis, we can see how the function behaves as it approaches 0.
When graphed, we observe that as ( x ) approaches 0 from both directions, the function indeed approaches the value of 7.5, confirming our conclusion about the limit.
In summary, the limit of the function ( f(x) ) as ( x ) approaches 0 is confirmed to be 7.5, supported by both numerical values and graphical analysis.
Calculate the values of ( f(x) = frac{3 sin(5x)}{sin(2x)} ) for ( x ) values even closer to 0 than those provided in the article. Use a calculator or software to find ( f(x) ) at ( x = pm 0.0001 ) and ( x = pm 0.00001 ). Compare these values with the ones given in the article and discuss your findings with your peers.
Use graphing software or an online graphing tool to plot the function ( f(x) = frac{3 sin(5x)}{sin(2x)} ) over the interval ([-0.1, 0.1]). Adjust the scale to clearly observe the behavior as ( x ) approaches 0. Share your graph with the class and explain how it supports the conclusion that the limit is 7.5.
Engage in a group discussion to explore the theoretical basis for the limit of the function. Use L’Hôpital’s Rule or Taylor series expansion to analytically prove that (lim_{x to 0} f(x) = 7.5). Present your proof to the class and discuss any assumptions or approximations made.
Research a real-world scenario where understanding the limit of a function is crucial. Prepare a short presentation on how limits are applied in fields such as engineering, physics, or economics. Highlight the importance of accurately determining limits in practical applications.
Create an interactive quiz using an online platform to test your understanding of limits. Include questions about the behavior of functions as they approach a point, the use of graphical analysis, and the application of L’Hôpital’s Rule. Share the quiz with your classmates and discuss the results.
Limit – The value that a function or sequence “approaches” as the input or index approaches some value. – As x approaches 0, the limit of sin(x)/x is 1.
Function – A relation between a set of inputs and a set of permissible outputs, typically defined by a mathematical expression. – The function f(x) = x^2 describes a parabola opening upwards.
Approaches – The process of getting closer to a particular value or condition, often used in the context of limits. – As x approaches infinity, the function f(x) = 1/x approaches 0.
Values – The numerical quantities assigned to variables or expressions, often representing the output of a function. – The values of the function f(x) = x^2 at x = 1, 2, and 3 are 1, 4, and 9, respectively.
Negative – Less than zero, often used to describe the sign of a number or the direction of a graph. – The function f(x) = -x^2 has a negative leading coefficient, causing the parabola to open downwards.
Positive – Greater than zero, often used to describe the sign of a number or the direction of a graph. – The derivative of f(x) = x^3 is positive for x > 0, indicating that the function is increasing in that interval.
Graph – A visual representation of a function or relation, typically in a coordinate plane. – The graph of the sine function oscillates between -1 and 1.
Behavior – The manner in which a function acts or changes, often analyzed as x approaches certain values. – The behavior of the function f(x) = 1/x near x = 0 is characterized by a vertical asymptote.
Converge – The process of approaching a limit or a common value, often used in the context of sequences or series. – The series ∑(1/n^2) converges to π^2/6.
Calculus – A branch of mathematics that studies continuous change, encompassing derivatives, integrals, limits, and infinite series. – Calculus is essential for understanding the dynamics of systems in physics and engineering.
