Finding Limits Numerically

## Description

Here, we will use a For loop to find the following limit numerically:

(1)
\begin{align} \lim_{x\to 8} e^{1/{(8 - x)}} \end{align}

## Sage Cell

Essentially what we're doing here is using the loop to print the values of $f(x)$ as it approaches x. We do this by looping i from 0 to 9 (10 is excluded) and using i as a negative exponent if 10, then subtracting that from 8. This allows us to approximate the limit, as we can see $f(x)$ exponentially growing; when i = 9, $f(8 - 10^{-i})$}} is approximately [[$8 \cdot 10^{434,294,481}$, an obscenely large number that allows us to approximate the left-side limit as $\infty$. To calculate the right-side limit, we simply add 10^(-i) to 8 instead of subtract it.

Here we can judge that from the right side, the function approaches $-\infty$, meaning that the limit as a whole does not exist.

#### Code

f(x) = exp(1/(8 - x))
for i in range (0, 10):
print "a =",
print (8 - 10^(-i)).n()
print "f(a) =",
print (f(8 - 10^(-i))).n()

f(x) = exp(1/(8 - x))
for i in range (0, 10):
print "a =",
print (8 + 10^(-i)).n()
print "f(a) =",
print (f(8 + 10^(-i))).n()


none

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