Evaluate the following limits :
`Lim_(x to sqrt(2))(x^(4)-4)/(x^(2)+3xsqrt(2)-8)`

Answer

To evaluate the limit

\[ \lim_{x \to \sqrt{2}} \frac{x^4 - 4}{x^2 + 3x\sqrt{2} - 8}, \]

we will follow these steps:

### Step 1: Substitute the limit value
First, let's substitute \( x = \sqrt{2} \) into the expression to check if we get a determinate form.

\[ \text{Numerator: } (\sqrt{2})^4 - 4 = 4 - 4 = 0. \]
\[ \text{Denominator: } (\sqrt{2})^2 + 3(\sqrt{2})(\sqrt{2}) - 8 = 2 + 3(2) - 8 = 2 + 6 - 8 = 0. \]

Since both the numerator and denominator approach 0, we have an indeterminate form \( \frac{0}{0} \).

### Step 2: Factor the numerator
We can factor the numerator \( x^4 - 4 \) using the difference of squares:

\[ x^4 - 4 = (x^2 - 2)(x^2 + 2). \]

### Step 3: Factor the denominator
Next, we need to factor the denominator \( x^2 + 3x\sqrt{2} - 8 \). We will look for two numbers that multiply to \(-8\) (the constant term) and add to \(3\sqrt{2}\) (the coefficient of \(x\)).

The expression can be factored as:

\[ x^2 + 3x\sqrt{2} - 8 = (x - \sqrt{2})(x + 4\sqrt{2}). \]

### Step 4: Rewrite the limit
Now we can rewrite the limit using the factored forms:

\[ \lim_{x \to \sqrt{2}} \frac{(x^2 - 2)(x^2 + 2)}{(x - \sqrt{2})(x + 4\sqrt{2})}. \]

### Step 5: Factor \(x^2 - 2\)
Notice that \(x^2 - 2\) can be factored further:

\[ x^2 - 2 = (x - \sqrt{2})(x + \sqrt{2}). \]

### Step 6: Cancel common factors
Now we can cancel the common factor \(x - \sqrt{2}\) from the numerator and denominator:

\[ \lim_{x \to \sqrt{2}} \frac{(x - \sqrt{2})(x + \sqrt{2})(x^2 + 2)}{(x - \sqrt{2})(x + 4\sqrt{2})} = \lim_{x \to \sqrt{2}} \frac{(x + \sqrt{2})(x^2 + 2)}{(x + 4\sqrt{2})}. \]

### Step 7: Substitute \(x = \sqrt{2}\) again
Now we can substitute \(x = \sqrt{2}\):

\[ \frac{(\sqrt{2} + \sqrt{2})(\sqrt{2}^2 + 2)}{(\sqrt{2} + 4\sqrt{2})} = \frac{(2\sqrt{2})(2 + 2)}{(5\sqrt{2})} = \frac{(2\sqrt{2})(4)}{(5\sqrt{2})}. \]

### Step 8: Simplify the expression
The \(\sqrt{2}\) in the numerator and denominator cancels out:

\[ = \frac{8}{5}. \]

Thus, the limit evaluates to:

\[ \boxed{\frac{8}{5}}. \]