Exponential Function Limits

Oct 13, 2025 by talynt.org 69 views

Introduction

In calculus, limits are used to study the behavior of functions as the input values approach a specific point. In this article, we will evaluate the limits of the function $f(x) = 6 \cos \left(\frac{1}{-4x - 8}\right)$ as $x$ approaches infinity and negative infinity.

The Function

The given function is $f(x) = 6 \cos \left(\frac{1}{-4x - 8}\right)$ . This function involves a trigonometric function, specifically the cosine function, and a rational function in the argument of the cosine function.

Evaluating the Limit as $x$ Approaches Infinity

To evaluate the limit of $f(x)$ as $x$ approaches infinity, we need to consider the behavior of the function as $x$ becomes very large.

\lim_{x \to \infty} f(x) = \lim_{x \to \infty} 6 \cos \left(\frac{1}{-4x - 8}\right)

As $x$ approaches infinity, the term $-4x - 8$ becomes very large and negative. The cosine function oscillates between $-1$ and $1$ as its argument changes. However, as the argument of the cosine function approaches $0$ , the cosine function approaches $1$ .

\lim_{x \to \infty} \frac{1}{-4x - 8} = 0

Therefore, as $x$ approaches infinity, the argument of the cosine function approaches $0$ , and the cosine function approaches $1$ .

\lim_{x \to \infty} f(x) = 6 \cos (0) = 6

Evaluating the Limit as $x$ Approaches Negative Infinity

To evaluate the limit of $f(x)$ as $x$ approaches negative infinity, we need to consider the behavior of the function as $x$ becomes very large and negative.

\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} 6 \cos \left(\frac{1}{-4x - 8}\right)

As $x$ approaches negative infinity, the term $-4x - 8$ becomes very large and positive. The cosine function oscillates between $-1$ and $1$ as its argument changes. However, as the argument of the cosine function approaches $0$ , the cosine function approaches $1$ .

\lim_{x \to -\infty} \frac{1}{-4x - 8} = 0

Therefore, as $x$ approaches negative infinity, the argument of the cosine function approaches $0$ , and the cosine function approaches $1$ .

\lim_{x \to -\infty} f(x) = 6 \cos (0) = 6

Conclusion

In this article, we evaluated the limits of the function $f(x) = 6 \cos \left(\frac{1}{-4x - 8}\right)$ as $x$ approaches infinity and negative infinity. We found that both limits approach $6$ .

Limit Theorems

The limit theorems are a set of rules that help us evaluate limits of functions. The following are some of the most commonly used limit theorems:

Sum Rule: $\lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
Product Rule: $\lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
Chain Rule: $\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x))$

Trigonometric Limits

The following are some of the most commonly used trigonometric limits:

$\lim_{x \to \infty} \sin x = 0$
$\lim_{x \to \infty} \cos x = 1$
$\lim_{x \to \infty} \tan x = 0$

Rational Function Limits

The following are some of the most commonly used rational function limits:

$\lim_{x \to \infty} \frac{1}{x} = 0$
$\lim_{x \to \infty} \frac{1}{x^2} = 0$
$\lim_{x \to \infty} \frac{1}{x^3} = 0$

Exponential Function Limits

The following are some of the most commonly used exponential function limits:

$\lim_{x \to \infty} e^x = \infty$
$\lim_{x \to -\infty} e^x = 0$
$\lim_{x \to \infty} e^{-x} = 0$

Logarithmic Function Limits

The following are some of the most commonly used logarithmic function limits:

$\lim_{x \to \infty} \ln x = \infty$
$\lim_{x \to 0^+} \ln x = -\infty$
$\lim_{x \to 0^-} \ln x = \infty$

Hyperbolic Function Limits

The following are some of the most commonly used hyperbolic function limits:

$\lim_{x \to \infty} \sinh x = \infty$
$\lim_{x \to -\infty} \sinh x = -\infty$
$\lim_{x \to \infty} \cosh x = \infty$
$\lim_{x \to -\infty} \cosh x = \infty$

Inverse Trigonometric Function Limits

The following are some of the most commonly used inverse trigonometric function limits:

$\lim_{x \to \infty} \arcsin x = \frac{\pi}{2}$
$\lim_{x \to -\infty} \arcsin x = -\frac{\pi}{2}$
$\lim_{x \to \infty} \arccos x = 0$
$\lim_{x \to -\infty} \arccos x = \pi$
$\lim_{x \to \infty} \arctan x = \frac{\pi}{2}$
$\lim_{x \to -\infty} \arctan x = -\frac{\pi}{2}$
Q&A: Evaluating Limits of Trigonometric Functions =====================================================

Q: What is the limit of $f(x) = 6 \cos \left(\frac{1}{-4x - 8}\right)$ as $x$ approaches infinity?

A: The limit of $f(x) = 6 \cos \left(\frac{1}{-4x - 8}\right)$ as $x$ approaches infinity is $6$ . This is because as $x$ approaches infinity, the term $-4x - 8$ becomes very large and negative, and the cosine function approaches $1$ .

Q: What is the limit of $f(x) = 6 \cos \left(\frac{1}{-4x - 8}\right)$ as $x$ approaches negative infinity?

A: The limit of $f(x) = 6 \cos \left(\frac{1}{-4x - 8}\right)$ as $x$ approaches negative infinity is also $6$ . This is because as $x$ approaches negative infinity, the term $-4x - 8$ becomes very large and positive, and the cosine function approaches $1$ .

Q: How do I evaluate the limit of a trigonometric function?

A: To evaluate the limit of a trigonometric function, you need to consider the behavior of the function as the input values approach a specific point. You can use the following steps:

Simplify the function by combining like terms and canceling out any common factors.
Use the trigonometric identities to rewrite the function in a simpler form.
Evaluate the limit by substituting the value of the input variable into the simplified function.

Q: What are some common trigonometric limits?

A: Some common trigonometric limits include:

$\lim_{x \to \infty} \sin x = 0$
$\lim_{x \to \infty} \cos x = 1$
$\lim_{x \to \infty} \tan x = 0$

Q: How do I evaluate the limit of a rational function?

A: To evaluate the limit of a rational function, you need to consider the behavior of the function as the input values approach a specific point. You can use the following steps:

Simplify the function by combining like terms and canceling out any common factors.
Use the rational function identities to rewrite the function in a simpler form.
Evaluate the limit by substituting the value of the input variable into the simplified function.

Q: What are some common rational function limits?

A: Some common rational function limits include:

$\lim_{x \to \infty} \frac{1}{x} = 0$
$\lim_{x \to \infty} \frac{1}{x^2} = 0$
$\lim_{x \to \infty} \frac{1}{x^3} = 0$

Q: How do I evaluate the limit of an exponential function?

A: To evaluate the limit of an exponential function, you need to consider the behavior of the function as the input values approach a specific point. You can use the following steps:

Simplify the function by combining like terms and canceling out any common factors.
Use the exponential function identities to rewrite the function in a simpler form.
Evaluate the limit by substituting the value of the input variable into the simplified function.

Q: What are some common exponential function limits?

A: Some common exponential function limits include:

$\lim_{x \to \infty} e^x = \infty$
$\lim_{x \to -\infty} e^x = 0$
$\lim_{x \to \infty} e^{-x} = 0$

Q: How do I evaluate the limit of a logarithmic function?

A: To evaluate the limit of a logarithmic function, you need to consider the behavior of the function as the input values approach a specific point. You can use the following steps:

Simplify the function by combining like terms and canceling out any common factors.
Use the logarithmic function identities to rewrite the function in a simpler form.
Evaluate the limit by substituting the value of the input variable into the simplified function.

Q: What are some common logarithmic function limits?

A: Some common logarithmic function limits include:

$\lim_{x \to \infty} \ln x = \infty$
$\lim_{x \to 0^+} \ln x = -\infty$
$\lim_{x \to 0^-} \ln x = \infty$

Q: How do I evaluate the limit of a hyperbolic function?

A: To evaluate the limit of a hyperbolic function, you need to consider the behavior of the function as the input values approach a specific point. You can use the following steps:

Simplify the function by combining like terms and canceling out any common factors.
Use the hyperbolic function identities to rewrite the function in a simpler form.
Evaluate the limit by substituting the value of the input variable into the simplified function.

Q: What are some common hyperbolic function limits?

A: Some common hyperbolic function limits include:

$\lim_{x \to \infty} \sinh x = \infty$
$\lim_{x \to -\infty} \sinh x = -\infty$
$\lim_{x \to \infty} \cosh x = \infty$
$\lim_{x \to -\infty} \cosh x = \infty$

Q: How do I evaluate the limit of an inverse trigonometric function?

A: To evaluate the limit of an inverse trigonometric function, you need to consider the behavior of the function as the input values approach a specific point. You can use the following steps:

Simplify the function by combining like terms and canceling out any common factors.
Use the inverse trigonometric function identities to rewrite the function in a simpler form.
Evaluate the limit by substituting the value of the input variable into the simplified function.

Q: What are some common inverse trigonometric function limits?

A: Some common inverse trigonometric function limits include:

$\lim_{x \to \infty} \arcsin x = \frac{\pi}{2}$
$\lim_{x \to -\infty} \arcsin x = -\frac{\pi}{2}$
$\lim_{x \to \infty} \arccos x = 0$
$\lim_{x \to -\infty} \arccos x = \pi$
$\lim_{x \to \infty} \arctan x = \frac{\pi}{2}$
$\lim_{x \to -\infty} \arctan x = -\frac{\pi}{2}$