What is the horizontal asymptote of $$ f(x)=6-e^{-x} $$ ? $$ y=\square $$

Question

UpStudy AI · Accepted Answer

1. We are given the function  
   $$
   f(x)=6-e^{-x}
   $$

2. To find the horizontal asymptote, we examine the limit of $$ f(x) $$ as $$ x $$ approaches infinity:  
   $$
   \lim_{x \to \infty} f(x)=\lim_{x \to \infty}\left(6-e^{-x}\right)
   $$

3. Recall that as $$ x $$ approaches infinity, the exponential term $$ e^{-x} $$ tends to $$ 0 $$ because the exponent $$-x$$ tends to $$-\infty$$. Thus,  
   $$
   \lim_{x \to \infty} e^{-x}=0
   $$

4. Applying this limit, we find:  
   $$
   \lim_{x \to \infty} f(x)=6-0=6
   $$

5. Therefore, the horizontal asymptote is given by  
   $$
   y=6
   $$

The final answer is:  
$$ y=6 $$
The horizontal asymptote is $$ y=6 $$.

What is the horizontal asymptote of ?

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Calculus Questions