Evaluate lim x → ∞ f ( x ) {\displaystyle\lim_{x\to\infty}{f(x)}} x → ∞ lim f ( x ) and lim x → − ∞ f ( x ) {\displaystyle\lim_{x\to-\infty}{f(x)}} x → − ∞ lim f ( x ) . f ( x ) = 1 − e − 2 x f\left(x\right)=1-e^{-2x}

Welcome back, everyone. Determine limit of FFX as X approaches positive or negative infinity for the function F of X equals 2 + E to the power of -3 X. Let's begin with the first limit. It says limit as X approaches positive infinity of FF X and F of X is 2 + E to the power of -3 X. And we begin with direct substitution. We end up with 2 + each the power of negative infinity. -3 multiplied by infinity gives us negative infinity, right? So this is an attempt to calculation. We can use direct substitution even when it's infinity. And now let's see what we are going to get. That's 2 plus, and whenever we raise a number to a negative exponent, we can rewrite it as 1 divided by. That number raises to a positive exponent, so 1 divided by e to the power of infinity, and essentially this term approaches 0 because we have 1 divided by infinity, and we can say that overall we end up with 2 + 1 divided by infinity. 1 divided by infinity approaches 0, so we're going to continue. 2 + 0, this gives us a limit of 2. So essentially, as X gets larger and larger, E to the power of -3 X gets closer and closer to zero, so we have a limit of 2. And now for the other limit, limits X approaches negative infinity of that same function 2 + E to the power of -3 X. Applying the direct substitution, we're going to get. 2 plus e to the power of negative negative infinity, right, because now we x approaches negative infinity and negative of negative infinity gives us positive infinity. So we end up with 2 plus e to the power of infinity and e to the power of infinity approaches infinity. 2 plus infinity essentially approaches infinity. So as X gets more and more negative, the exponential term gets closer and closer to a very large number which we define as infinity. 2 has no effect, right? 2 plus infinity gives us a limit of infinity, and that's it. We have our answers. The first limit as X approaches infinity is 2, and the second one is X approaches negative infinity is infinity. Thank you for watching.

1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

2:49 minutes

Problem 2.R57

Textbook Question

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

$f (x) = 1 - e^{- 2 x}$

Verified step by step guidance

Identify the function given: $ f(x) = 1 - e^{-2x} $. We need to evaluate the limits as $ x \to \infty $ and $ x \to -\infty $.

Consider the limit $ \lim_{x \to \infty} f(x) $. As $ x \to \infty $, the term $ e^{-2x} $ approaches 0 because the exponent $-2x$ becomes very large and negative, making $ e^{-2x} $ very small.

Thus, $ \lim_{x \to \infty} f(x) = 1 - 0 = 1 $.

Now, consider the limit $ \lim_{x \to -\infty} f(x) $. As $ x \to -\infty $, the term $ e^{-2x} $ approaches infinity because the exponent $-2x$ becomes very large and positive.

Therefore, $ \lim_{x \to -\infty} f(x) = 1 - \infty = -\infty $.

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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 describe the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, allowing us to determine horizontal asymptotes and the end behavior of functions. Evaluating limits at infinity often involves simplifying the function to identify dominant terms that dictate its behavior.

Exponential Functions

Exponential functions, such as f(x) = 1 - e^(-2x), are characterized by a constant base raised to a variable exponent. These functions exhibit rapid growth or decay, depending on the sign of the exponent. Understanding their properties, including their limits as x approaches infinity or negative infinity, is essential for analyzing their long-term behavior and applications in various fields.

Continuous Functions

A function is continuous if there are no breaks, jumps, or holes in its graph. This property is important when evaluating limits, as continuous functions allow for the direct substitution of values. In the context of limits at infinity, continuity ensures that the limit can be determined by examining the function's behavior without encountering undefined points.

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