Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.35

Chapter 2, Problem 2.35

Determine the following limits.

lim x→∞ sin x / e^x

Verified step by step guidance

Step 1: Understand the problem. We need to find the limit of the function $ \frac{\sin x}{e^x} $ as $ x $ approaches infinity.

Step 2: Analyze the behavior of the numerator and the denominator separately. The function $ \sin x $ oscillates between -1 and 1 for all $ x $.

Step 3: Consider the behavior of the denominator $ e^x $. As $ x $ approaches infinity, $ e^x $ grows exponentially and becomes very large.

Step 4: Apply the Squeeze Theorem. Since $ -1 \leq \sin x \leq 1 $, we have $ -\frac{1}{e^x} \leq \frac{\sin x}{e^x} \leq \frac{1}{e^x} $.

Step 5: Evaluate the limits of the bounding functions. As $ x \to \infty $, both $ \frac{1}{e^x} \to 0 $ and $ -\frac{1}{e^x} \to 0 $. By the Squeeze Theorem, $ \lim_{x \to \infty} \frac{\sin x}{e^x} = 0 $.

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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 input approaches infinity. In this context, we analyze how the function behaves as x becomes very large, which can reveal whether the function approaches a specific value, diverges, or oscillates.

Behavior of Sinusoidal Functions

The sine function oscillates between -1 and 1 for all real numbers. This bounded behavior is crucial when evaluating limits involving sine, as it indicates that despite the oscillation, the overall contribution of sin(x) becomes negligible compared to other functions that grow without bound, such as exponential functions.

Exponential Growth

Exponential functions, like e^x, grow significantly faster than polynomial or sinusoidal functions as x approaches infinity. This rapid growth is key in limit problems, as it often leads to the conclusion that terms involving e^x will dominate the behavior of the limit, driving the overall limit towards zero when combined with bounded functions.

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Exponential Functions

Related Practice

Textbook Question

Suppose f(x)→100 and g(x)→0, with g(x)<0 as x→2. Determine lim x→2 f(x) / g(x).

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Textbook Question

If a function f represents a system that varies in time, the existence of lim ${$\displaystyle\[\lim$_{t$\rightarrow\]\infty$}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a culture of tumor cells is given by $p (t) = \frac{3500 t}{t + 1}$ .

Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2−25 / x−5; a=5

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Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {x^2+1 if x≤−1

3 if x>−1; a=−1

Textbook Question

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?

f(x)=(2x−3)^2/3

Textbook Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.

$f (x) = 1 - \ln x$

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Determine the following limits. lim x→∞ sin x / e^x

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Key Concepts

Limits at Infinity

Behavior of Sinusoidal Functions

Exponential Growth

Determine the following limits.

lim x→∞ sin x / e^x