Determine the following limits. lim t → ∞ cos ( t ) e 3 t {\displaystyle\lim_{t\to\infty}\frac{\cos\left(t\right)}{e^{3t}}}

Welcome back, everyone. Find the limit. Limit as he approaches infinity of sinet divided by E to the power of T T, and we're given for answer choices A 0 B infinity, C1 and D negative infinity. So, let's analyze the given limit. Limit as T approaches infinity. We have sin T in the numerator, and in the denominator, we have an exponential function E to the power of TT. Now, we're going to apply the right substitution and also apply the analytical method. We have a rational function that consists of two functions, and one of them is sine T in the numerator. And we know that it keeps oscillating between -1 and 1, right? It is a periodic function, so we have Y of X as a function and sine T as T approaches infinity. It just keeps oscillating within the range between -1 and 1. On average, that's basically 0. Now, what about the denominator? We have an exponential function, so if we plot it. We have to recall that as E increases. E to T goes to infinity, right? So basically an exponential function grows infinitely. So as T goes to infinity, the denominator grows towards infinity. And now when we look at this attempted calculation, we are dividing a number that oscillates between 1 and 1 by an infinitely large number. So this means that the limit to this specific function is going to be equal to 0. And our final answer to the problem corresponds to the answer choice A. Thank you for watching.

Table of contents

Skip topic navigation

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

Previous problemNext problem

2:04 minutes

Problem 2.50

Textbook Question

Determine the following limits.
$\lim_{t \to \infty} \frac{\cos (t)}{e^{3 t}}$

Verified step by step guidance

Identify the limit expression: $ \lim_{t \to \infty} \frac{\cos(t)}{e^{3t}} $.

Recognize that $ \cos(t) $ oscillates between -1 and 1 for all $ t $.

Observe that $ e^{3t} $ grows exponentially as $ t \to \infty $.

Consider the behavior of the fraction: as the denominator $ e^{3t} $ becomes very large, the fraction $ \frac{\cos(t)}{e^{3t}} $ approaches zero.

Conclude that the limit is zero because the exponential growth in the denominator dominates the bounded oscillation of the numerator.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the variable approaches infinity. In this context, we analyze how the function behaves when 't' becomes very large. Understanding this concept is crucial for determining whether the limit converges to a finite value, diverges, or approaches zero.

Behavior of Exponential Functions

Exponential functions, such as e^(3t), grow significantly faster than trigonometric functions like cosine(t) as 't' approaches infinity. This rapid growth means that even though cosine oscillates between -1 and 1, the denominator will dominate the fraction, leading to a limit of zero. Recognizing this behavior is essential for solving the limit problem.

Trigonometric Functions and Boundedness

Trigonometric functions, such as cosine, are bounded, meaning they have a fixed range of values (between -1 and 1). This property is important when evaluating limits because it indicates that the numerator will not grow indefinitely, allowing us to compare it effectively with the rapidly increasing denominator in the limit expression.

Watch next

Master Finding Limits by Direct Substitution with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Evaluate each limit.

lim x→0 cos x−1 / sin^2x

Textbook Question

Evaluate each limit.

lim x→0 e^4x−1 / e^x−1

Textbook Question

Evaluate each limit.

lim x→0+ 1−cos^2x / sin x

Textbook Question

Evaluate each limit.

lim x→e^2 ln^2x−5 ln x+6 lnx−2

Textbook Question

Evaluate $\lim_{x \to \infty} f (x)$ and $\lim_{x \to - \infty} f (x)$ .

$f (x) = \frac{4 x^{3} + 1}{1 - x^{3}}$

Textbook Question

Evaluate ${\displaystyle\lim_{x\to\infty}{f(x)}}$ and ${\displaystyle\lim_{x\to-\infty}{f(x)}}$ .

$f (x) = \frac{6 e^{x} + 20}{3 e^{x} + 4}$

Textbook Question

Complete the following steps for the given functions.

c. Graph $f$ and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.

$f (x) = \frac{x^{2} - 3}{x + 6}$

Textbook Question

Complete the following steps for the given functions.

c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.

$f (x) = \frac{4 x^{3} + 4 x^{2} + 7 x + 4}{x^{2} + 1}$

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap

Determine the following limits.limt→∞cos(t)e3t{\displaystyle\lim_{t\to\infty}\frac{\cos\left(t\right)}{e^{3t}}}

Key Concepts

Limits at Infinity

Behavior of Exponential Functions

Trigonometric Functions and Boundedness

Watch next

Determine the following limits.
$\lim_{t \to \infty} \frac{\cos (t)}{e^{3 t}}$