Question

Evaluate the following limits or state that they do not exist. $$\lim _{t \rightarrow \infty} \frac{\cos t}{e^{3 t}}$$

Short answer

Answer: The limit of the function \(\frac{\cos t}{e^{3t}}\) as t approaches infinity is 0.

Step-by-step solution

Analyze the behavior of the numerator and denominator as t approaches infinity.

As \(t \rightarrow \infty\), the cosine function oscillates between -1 and 1 (\(-1 \leq \cos t \leq 1\)). And we also know that as \(t \rightarrow \infty\), the exponential function \(e^{3t}\) grows rapidly and without bound.

Determine the limit.

In this case, as the numerator oscillates between -1 and 1, the denominator is growing rapidly towards positive infinity. Hence, we can say that the fraction gets smaller and ultimately converges to 0. Therefore, the limit exists and is equal to 0. The limit can be represented as follows: $$\lim_{t \rightarrow \infty}\frac{\cos t}{e^{3t}}=0$$

Key concepts

Infinite Limits

When students approach the concept of infinite limits, it is indispensable to view it as an exploration of a function's behavior as its argument moves towards infinity or negative infinity. In the mathematical scenario where the function's independent variable, represented by 't' in the above exercise, trends infinitely large, the general question is: What happens to the function's value?

In our function \( \frac{\cos t}{e^{3t}} \), we see that while the numerator, \( \cos t \), remains bounded between -1 and 1, the denominator—an exponential function—grows so rapidly that it dwarfs the numerator's oscillations. Since the numerator oscillates within a fixed range and the denominator becomes exceedingly large, the ratio tends to zero. The 'infinite aspect' comes from considering what occurs as 't' goes to infinity. It's important to note that not all functions tend toward a finite number, some approach infinity themselves, or do not have a limit at all. This is the essence of understanding infinite limits: discerning the behavior of mathematical expressions as they stretch out towards or into the infinite.

Exponential Functions

Exponential functions paint a picture of rapid growth or decay, depending on the form they take. The classic form \( e^x \) embodies growth that is continually accelerating—it increases faster as 'x' increases. In calculus, these functions are central to discussions on limits, differentiation, and integration due to their unique properties.

For example, in the textbook exercise, we encounter \( e^{3t} \), which shows exponential growth with '3t' in the exponent. This indicates that for every increment in 't', the function's value doesn’t just increase but multiplies by a factor much greater than linear growth. An important property of exponential functions—responsible for their steep ascent—is that their rate of growth is proportional to their current value. Therefore, as \( t \rightarrow \)infinity, \( e^{3t} \) quickly eclipses any bounded function, resulting in the limit tending towards zero when the exponential function is in the denominator.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic in nature; they repeat their values in a predictable pattern. This aspect is fundamental to their identity and deeply influences their behavior in limits and calculus at large. Analyzing our exercise, \( \cos t \) is bound to repeat its values between -1 and 1 no matter the value of 't'.

Even as 't' goes to infinity, cosine won't break out of this cycle—it is forever oscillating. This makes the function incredibly predictable in long-term behavior. When dealing with limits and the interaction of trigonometric functions with others, like exponential functions, this predictable behavior can greatly simplify finding limits. The bounded nature of trigonometric functions such as cosine and sine is a stark contrast to the unbounded growth or decay of exponential functions and is what ultimately leads to the result of the limit in the given exercise being zero.