Suppose that
lim t→1 (f(t) − e^π)/ (t − 1) = 16.
What is limit→1 f(t)?
I don't understand, is f(1) = e^x? but how do I show it? am I allowed to rearrange that equation to f(t) = 16(t-1) +e^π inside of a limit? ughhhh I dont understand, any help is appreciated!
what is the limit t→1 f(t)?**
note that at t=1, we have (f(1)-e^π)/(1-1) = 16
But division by zero is undefined, and the only way that limit can be 16 is if
f(1) - e^π is also zero.
So, lim(t->1) f(t) = e^π
thank you!!
To find the limit of f(t) as t approaches 1, we have the given expression:
lim t→1 (f(t) - e^π) / (t - 1) = 16.
Since we want to determine lim t→1 f(t), one approach is to rearrange the equation to isolate f(t). However, we need to be cautious when manipulating limits. It is essential to ensure that any rearrangements made are mathematically valid.
Here's one way to approach it:
Start with the given equation:
lim t→1 (f(t) - e^π) / (t - 1) = 16.
We want to find lim t→1 f(t), so let's multiply both sides of the equation by (t - 1):
(t - 1) * lim t→1 (f(t) - e^π) / (t - 1) = (t - 1) * 16.
The (t - 1) terms cancel on the left side, giving us:
lim t→1 (f(t) - e^π) = 16(t - 1).
Now, let's rearrange our equation to isolate f(t):
lim t→1 f(t) - lim t→1 e^π = 16(t - 1).
Since t approaches 1, we can say:
lim t→1 e^π = e^π.
Substituting this back into our equation:
lim t→1 f(t) - e^π = 16(t - 1).
Finally, to find lim t→1 f(t), we can add e^π to both sides:
lim t→1 f(t) = 16(t - 1) + e^π.
Therefore, the limit of f(t) as t approaches 1 is given by the expression:
lim t→1 f(t) = 16(t - 1) + e^π.