find the limit of f'(x) = 1/(√x) using the limit definition of derivative.
what is x approaching?
x is approaching 0
To find the limit of f'(x) = 1/√x using the limit definition of the derivative, we first need to recall the definition of the derivative:
f'(x) = lim(h→0) [f(x+h) - f(x)] / h
In this case, we have f'(x) = 1/√x. So, we need to substitute this expression into the definition and evaluate the limit as h approaches 0.
f'(x) = lim(h→0) [1/√(x+h) - 1/√x] / h
Now, let's simplify this expression step by step:
Step 1: Rationalize the denominator.
f'(x) = lim(h→0) [√(x+h) - √x] / (h * √(x+h) * √x)
Step 2: Multiply both the numerator and denominator by conjugate of the numerator.
f'(x) = lim(h→0) [√(x+h) - √x] * (√(x+h) + √x) / (h * √(x+h) * √x) * (√(x+h) + √x)
Step 3: Apply the difference of squares formula.
f'(x) = lim(h→0) [√(x+h)² - √x²] / (h * √(x+h) * √x) * (√(x+h) + √x)
= lim(h→0) (x + h - x) / (h * √(x+h) * √x) * (√(x+h) + √x)
= lim(h→0) h / (h * √(x+h) * √x) * (√(x+h) + √x)
Step 4: Cancel out the h terms.
f'(x) = lim(h→0) 1 / (√(x+h) * √x) * (√(x+h) + √x)
Step 5: Factor out a common term.
f'(x) = lim(h→0) 1 / (√x * (√(x+h) * √x)) * (√(x+h) + √x)
Step 6: Simplify the expression inside the limit.
f'(x) = 1 / (√x * (√x)) * (√x + √x)
= 1 / (√x) * 2√x
= 2
So, the limit of f'(x) = 1/√x using the limit definition of the derivative is 2.