The expression : lim h--> 0 (√h+9 -3)/ h represents the slope of the tangent of some
function y = f(x) at x =4.
Determine the functions y = f(x)
Determine the numeric value of lim h--> 0 (√h+9 -3)/ h
y = f(x)
slope at x = [f(x+h) - f(x) ] /h as h -->0
What you typed makes no sense. Perhaps you mean
lim h--> 0 [ √ (h+9) - 3 ] / h at x = 4
looks like f(4) = 3
√ (h+9) = (9+h)^1/2 = 3 + (1/2)(1/3)h +(1/2)(-1/2)/2 *(1/2)^-1.5 h^2 ....
(binomial series)
so as h approaches 0
[ 3 + 1/6 h + ..... - 3 ] /h = 1/6 h /h = 1/6
sorry for the poor wording.
The expression : lim [√(h+9) -3]/ h as h --> 0 represents the slope of the tangent of some function f(x) when x = 4.
How do you determine the function f(x)?
Opps again. I meant to say:
Determine the function f(x)
To determine the functions y = f(x), we need to find the derivative of the function at x = 4.
The expression (√h+9 - 3)/h represents the definition of derivative. By taking the limit of this expression as h approaches 0, we can find the derivative.
To find the derivative, let's simplify the expression first:
(√h + 9 - 3) / h = (√h + 6) / h
Now, let's calculate the limit using this expression:
lim h --> 0 (√h + 6) / h
To evaluate this limit, we can use L'Hôpital's rule. This rule states that for a limit of the form (f(x) / g(x)) as x approaches a, if both f(x) and g(x) approach 0 or ±∞, we can calculate the limit by finding the derivative of the numerator and denominator and then taking the limit again.
Taking the derivative of the numerator and denominator, we get:
d/dh (√h + 6) = 1 / (2√h)
d/dh h = 1
Now, let's calculate the limit again using the derivatives:
lim h --> 0 (1 / (2√h)) / 1
Simplifying this expression, we get:
lim h --> 0 1 / (2√h)
As h approaches 0, the denominator (√h) approaches 0.
Therefore, the limit of (√h + 9 - 3) / h as h approaches 0 is undefined.
Since the limit is undefined, it means that the slope of the tangent of the function y = f(x) at x = 4 also does not exist.
To determine the function y = f(x), we would need more information or context about the relationship between x and y. Without any additional information, it is not possible to determine the specific function y = f(x).