Question

If $f(x)=\sqrt{10-x}$, then $\lim_{x\to 1}\frac{f(x)-f(1)}{x-1}$ is equal to:

• A. \(3\)
• B. \(\frac{1}{3}\)
• C. \(\frac{1}{6}\)
• D. \(-\frac{1}{6}\)
• E. \(\frac{3}{2}\)

Hint:

Limits of the form \(\frac{f(x)-f(a)}{x-a}\) directly represent \(f'(a)\). Use differentiation instead of expanding the limit manually.

Answer 1

The Correct Option is D

Step 1: Recognize the definition of derivative.
The given limit is:
\[ \lim_{x\to 1}\frac{f(x)-f(1)}{x-1} \] This is the definition of the derivative of \(f(x)\) at \(x=1\), i.e.,
\[ f'(1) \]

Step 2: Write the function clearly.
\[ f(x)=\sqrt{10-x}=(10-x)^{1/2} \]

Step 3: Differentiate \(f(x)\).
Using the chain rule:
\[ f'(x)=\frac{1}{2}(10-x)^{-1/2}\cdot(-1) \] \[ f'(x)=-\frac{1}{2\sqrt{10-x}} \]

Step 4: Evaluate the derivative at \(x=1\).
\[ f'(1)=-\frac{1}{2\sqrt{10-1}} \] \[ f'(1)=-\frac{1}{2\sqrt{9}} \] \[ f'(1)=-\frac{1}{2\times 3} \]

Step 5: Simplify the expression.
\[ f'(1)=-\frac{1}{6} \]

Step 6: Interpret the result.
Thus, the value of the given limit is equal to the derivative at \(x=1\), which is \(-\frac{1}{6}\).

Step 7: Final answer.
\[ \boxed{-\frac{1}{6}} \] which matches option \((4)\).