Lim tan(x+ pi/4)^1/x x-->0
as x--->0
tan (x+π/4) ---> tan π/4 = 1
so lim tan(x+π/4) /x ---> 1/0
which is undefined or infinity
To find the limit of the given function as x approaches 0, we can use the concept of limits and apply algebraic manipulations. Let's break down the steps to evaluate this limit.
1. Start with the given function: lim[x→0] tan((x + π/4)^(1/x)).
2. We can observe that as x approaches 0, the expression (x + π/4)^(1/x) becomes indeterminate because raising any non-zero number (x + π/4) to the power of 1/0 results in an undefined value.
3. To progress further, let's rewrite the given expression in a way that allows us to simplify it. Use the property that tan(π/4) = 1 to manipulate the expression:
lim[x→0] tan((x + π/4)^(1/x)) = lim[x→0] tan((x + π/4)^(1/x) * (1/π) * π/4).
4. Now, consider the term (1/x) * π/4 as x approaches 0. This term represents an infinity multiplied by a finite value. Thus, the product will be undefined.
5. Finally, evaluate the limit: lim[x→0] tan((x + π/4)^(1/x)) = undefined.
In summary, as x approaches 0, the given function does not have a finite limit; it is undefined.