The limit represents the derivative of some function f at some number a. Select an appropriate f(x) and a.
lim (cos(pi+h)+1)/h
h->0
answers are
f(x) = tan(x), a = pi
f(x) = cos(x), a = pi/4
f(x) = cos(x), a = pi
f(x) = sin(x), a = pi
Using La'Hopital's rule...
lim = d/dx numerator/d/dx denominator
= -sin(pi+h)(1)/1= -sinPI
So none of the answers are really right, however, one can note that sinPI=-sinPI=0, so the last answer is right because it is zero. However, one could have written just as well
f(x)= 40.3 sinPI and that would be right. And because of this, equalling zero, the first answer is just as right. The question is not well crafted, unless I missed something.
To find an appropriate function f(x) and number a such that the limit represents the derivative, we need to find a function that satisfies the definition of a derivative.
Recall that the derivative of a function f(x) at a point a is defined as the limit:
f'(a) = lim (f(x) - f(a))/(x - a)
x->a
In this case, we have the limit:
lim (cos(pi+h)+1)/h
h->0
To find a suitable f(x) and a, we can find the derivative of different functions and see which one matches the given limit.
Let's begin by considering f(x) = tan(x) and a = pi:
f'(a) = lim (tan(x) - tan(pi))/(x - pi)
x->pi
To simplify the expression, we can use the identity for the tangent difference:
tan(x) - tan(pi) = (tan(x) - tan(pi))/(1 + tan(x)tan(pi))
Applying this to the expression:
f'(a) = lim (tan(x) - tan(pi))/(x - pi) * (1 + tan(x)tan(pi))/(1 + tan(x)tan(pi))
x->pi
Expanding the numerator:
f'(a) = lim [tan(x) - tan(pi) + tan(x)tan(pi) - tan(pi)tan(x)] / [(x - pi)(1 + tan(x)tan(pi))]
x->pi
Now we can cancel out the tan(pi) terms:
f'(a) = lim [tan(x) - tan(x)] / [(x - pi)(1 + tan(x)tan(pi))]
x->pi
Simplifying further:
f'(a) = lim 0 / [(x - pi)(1 + tan(x)tan(pi))]
x->pi
Since the numerator is 0, the expression becomes:
f'(a) = lim 0 / (x - pi)
x->pi
At this point, we can see that the given limit matches the derivative of f(x) = tan(x) at a = pi:
lim (cos(pi+h)+1)/h = 0 / (x - pi)
h->0
Therefore, an appropriate f(x) = tan(x) and a = pi.
Note: The other options do not match the given limit exactly.