Question

Evaluate the given limits using the graph of the function. \(f(x)=x^{2} \sin (\pi x)\) (a) \(\lim _{x \rightarrow-\infty} f(x)\) (b) \(\lim _{x \rightarrow \infty} f(x)\)

Short answer

Both limits do not exist due to unbounded oscillation.

Step-by-step solution

Understand the Graph Behavior at Infinity

The function given is \(f(x)=x^{2} \sin(\pi x)\). We are tasked to find the behavior of this function as \(x\) approaches negative and positive infinity. Since \(x^2\) grows towards infinity for any non-zero values of \(x\), and \(\sin(\pi x)\) oscillates between -1 and 1, we need to analyze how these interact.

Analyzing Oscillation of Sine Function

Notice that \(\sin(\pi x)\) is bounded between -1 and 1, thus the output of the sine function doesn't grow. It only oscillates, contributing periodic multiplicand that increases with the magnitude of \(x^2\).

Evaluating Limit as x Approaches Negative Infinity

As \(x \rightarrow -\infty\), \(x^2 \rightarrow \infty\), but because \(\sin(\pi x)\) remains oscillating between -1 and 1, the overall product also oscillates vastly between \(-x^2\) and \(x^2\). Therefore, \(\lim_{x \rightarrow -\infty} f(x)\) does not settle to a fixed number.

Evaluating Limit as x Approaches Positive Infinity

Similarly, as \(x \rightarrow \infty\), \(x^2 \rightarrow \infty\) and \(\sin(\pi x)\) continues to oscillate between -1 and 1. Thus, the product \(x^2 \sin(\pi x)\) oscillates without an upper or lower bound, leading to an undefined limit.

Key concepts

Graph Behavior

When we analyze a graph to understand the behavior of a function at infinity, we are looking at what happens to the function as the input values become very large or very small. For the function \(f(x) = x^{2} \sin(\pi x)\), we can break it down into two important parts that shape its graph behavior at infinity:

• The term \(x^2\) is a parabola that opens upwards, meaning its value increases towards infinity as \(x\) becomes larger, whether in the positive or negative direction.
• In contrast, the \(\sin(\pi x)\) part is a periodic function that oscillates between -1 and 1, regardless of how large \(x\) becomes.

Combining these two behaviors, the graph of \(f(x)\) doesn't settle down to any particular value at infinity. Instead, it shows a pattern of growth due to \(x^2\) alongside relentless oscillation from \(\sin(\pi x)\). Thus, as \(x\) goes to infinity or negative infinity, the function exhibits an oscillating growth that prevents it from having a finite limit.

Oscillating Functions

Oscillating functions, like the sine function in \(\sin(\pi x)\), are functions that do not converge to a single limit but instead fluctuate within a definite range. For \(\sin(\pi x)\):

• The values are limited between -1 and 1, showing no tendency towards a single number.
• This periodic nature means, while it repeats every interval, for our problem, it's multiplied by the growing \(x^2\).

Oscillation impacts function behavior significantly. Even when multiplied by something growing, i.e., \(x^2\), the principle that the function continues to oscillate and has no fixed direction or terminal point remains. In our specific limit problem, the continuous oscillation leads to never-ending expansion between negative and positive peaks due to the oscillating \(\sin(\pi x)\). This oscillation makes clear that the limit does not stabilize at infinity.

Infinite Limit Analysis

To understand limits at infinity, we observe what occurs to \(f(x)\) as \(x\) becomes very large in magnitude, either positively or negatively. We look specifically at how \(x^{2} \sin(\pi x)\) behaves. Performing infinite limit analysis involves these considerations:

• The massive growth of \(x^2\): As \(x\) becomes larger, both negatively or positively, \(x^2\) increases without bounds.
• The bounded oscillation of \(\sin(\pi x)\): While \(x^2\) grows, \(\sin(\pi x)\) maintains its bounds, creating an oscillating influence.

Combining these effects means that despite \(x^2\) pushing the function towards infinity or negative infinity, \(\sin(\pi x)\) causes \(f(x)\) to continuously swing between large positive and negative values. The interaction of unbounded growth with bounded oscillation results in an undefined limit. Hence, \(\lim_{x \to \pm \infty} f(x)\) doesn't reach a single number, echoing the complex dance between explosive growth and structured oscillation.