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Chapter 2: Problem 4

Describe the end behavior of \(g(x)=e^{-2 x}\).

Short Answer

Expert verified
Based on the given function \(g(x) = e^{-2x}\), describe its end behavior as \(x\) approaches positive infinity and negative infinity. Answer: As \(x\) approaches positive infinity, \(g(x)\) approaches 0, and as \(x\) approaches negative infinity, \(g(x)\) approaches infinity.

Step by step solution

01

Find the limit as \(x\) approaches positive infinity

We need to find the limit of the function as \(x\) approaches positive infinity, which is represented mathematically as \(\lim_{x\to\infty} e^{-2x}\). To do this, we can simply observe that as \(x\) gets larger, the exponent \(-2x\) becomes more negative, causing the function value to approach zero. Therefore, the end behavior of the function as \(x\) approaches positive infinity is: $$\lim_{x\to\infty} e^{-2x} = 0$$
02

Find the limit as \(x\) approaches negative infinity

Now, we need to find the limit of the function as \(x\) approaches negative infinity, represented mathematically as \(\lim_{x\to-\infty} e^{-2x}\). In this case, as \(x\) gets more negative, the exponent \(-2x\) becomes more positive, causing the function value to approach infinity. Therefore, the end behavior of the function as \(x\) approaches negative infinity is: $$\lim_{x\to-\infty} e^{-2x} = \infty$$
03

Describe the end behavior

In conclusion, the end behavior of the function \(g(x) = e^{-2x}\) can be described as follows: As \(x\) approaches positive infinity, \(g(x)\) approaches 0, and as \(x\) approaches negative infinity, \(g(x)\) approaches infinity.

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