To solve the limit $\lim_{x \to 1} \frac{\sqrt{x} - 1 + \sqrt{x - 1}}{\sqrt{x^2 - 1}}$ using a simple method, let's follow the steps below:

Step 1: Evaluate the limit directly at $x = 1$

Substitute $x = 1$ into the expression:

$\frac{\sqrt{1} - 1 + \sqrt{1 - 1}}{\sqrt{1^2 - 1}} = \frac{1 - 1 + \sqrt{0}}{\sqrt{1 - 1}} = \frac{0 + 0}{0}$

This results in an indeterminate form $\frac{0}{0}$, so we need to simplify the expression further.

Step 2: Simplify the expression

We'll simplify the expression by first analyzing each component of the fraction.

Simplifying the numerator:

• $\sqrt{x} - 1$ is small when $x$ is close to 1.
• $\sqrt{x - 1}$ is also small when $x$ is close to 1.

So the numerator can be tricky to handle directly. Let's apply a simple technique to rewrite the numerator.

Step 3: Approximation and Substitution

We'll use first-order approximations for the square root functions around $x = 1$:

$\sqrt{x} \approx 1 + \frac{1}{2}(x - 1)$

$\sqrt{x - 1} \approx \sqrt{0 + h} \approx \sqrt{h}$

where $h = x - 1$.

Substitute these into the original expression:

$\lim_{x \to 1} \frac{(1 + \frac{1}{2}(x - 1)) - 1 + \sqrt{x - 1}}{\sqrt{x^2 - 1}}$

Simplifying the numerator:

$\lim_{x \to 1} \frac{\frac{1}{2}(x - 1) + \sqrt{x - 1}}{\sqrt{(x - 1)(x + 1)}}$

Step 4: Factor and Cancel Terms

The denominator can be rewritten as:

$\sqrt{x^2 - 1} = \sqrt{(x - 1)(x + 1)}$

So the limit becomes:

$\lim_{x \to 1} \frac{\frac{1}{2}(x - 1) + \sqrt{x - 1}}{\sqrt{(x - 1)(x + 1)}}$

Factor out $\sqrt{x - 1}$ from the numerator:

$\lim_{x \to 1} \frac{\sqrt{x - 1}\left(\frac{1}{2}\sqrt{x - 1} + 1\right)}{\sqrt{(x - 1)(x + 1)}}$

Simplifying further:

$\lim_{x \to 1} \frac{\sqrt{x - 1}}{\sqrt{x - 1}} \cdot \frac{\frac{1}{2}\sqrt{x - 1} + 1}{\sqrt{x + 1}}$

Since $\frac{\sqrt{x - 1}}{\sqrt{x - 1}} = 1$, the limit simplifies to:

$\lim_{x \to 1} \frac{\frac{1}{2}\sqrt{x - 1} + 1}{\sqrt{x + 1}}$

Step 5: Substitute $x = 1$

Now substitute $x = 1$ into the simplified expression:

$\frac{\frac{1}{2}\sqrt{0} + 1}{\sqrt{1 + 1}} = \frac{1}{\sqrt{2}}$

Thus, the limit is:

$\boxed{\frac{1}{\sqrt{2}}}$

Would you like further details or have any questions?

Follow-up Questions:

1. How do we handle indeterminate forms in limits?
2. What is the importance of the first-order approximation in solving limits?
3. Why is it helpful to factor terms when simplifying limits?
4. What does the expression $\sqrt{x^2 - 1}$ signify geometrically?
5. How can we apply the same method to more complex limits?
6. Why does substitution often lead to simpler forms in limits?
7. How does understanding square root approximations help in solving limits?
8. What are some other methods to evaluate indeterminate forms besides substitution?

Tip:

Always check if an expression can be factored or simplified before directly applying limit properties. This often simplifies the problem significantly!