Evaluate

Question

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

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

Accepted Answer

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.

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

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

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

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