The limit in the image is:

$\lim_{x \to -1} \left( x^2 \right)^{\frac{1}{1 - x^2}}.$

Solution Steps:

We analyze the limit step-by-step:

1. Simplify the base $x^2$:

   • As $x \to -1$, the square of $x$ is $x^2 \to (-1)^2 = 1$. Therefore, the base approaches $1$.

2. Analyze the exponent $\frac{1}{1 - x^2}$:

   • As $x \to -1$, $x^2 \to 1$, so $1 - x^2 \to 1 - 1 = 0$.
   • This makes the denominator $1 - x^2 \to 0$, which implies the exponent $\frac{1}{1 - x^2} \to \infty$ (positive infinity).

3. Form of the limit:

   • The expression now has the form: $1^{\infty}.$
   • This is an indeterminate form, so we take the natural logarithm to simplify.

Natural Logarithm Approach:

Let $L = \lim_{x \to -1} \left( x^2 \right)^{\frac{1}{1 - x^2}}$.

Take the natural logarithm on both sides: $\ln L = \lim_{x \to -1} \frac{\ln (x^2)}{1 - x^2}.$

Step-by-step Analysis of the Logarithm:

• Numerator: $\ln (x^2) = 2 \ln |x|$ because $\ln (a^b) = b \ln a$.

  • As $x \to -1$, $\ln |x| = \ln 1 = 0$, so $\ln (x^2) \to 0$.

• Denominator: $1 - x^2 \to 0$ as $x \to -1$.

Thus, we have a fraction of the form $\frac{0}{0}$, so we apply L'Hôpital's Rule.

Applying L'Hôpital's Rule:

Differentiate the numerator and denominator with respect to $x$:

• Derivative of the numerator $\ln (x^2)$: $\frac{d}{dx} \ln (x^2) = \frac{2}{x}.$

• Derivative of the denominator $1 - x^2$: $\frac{d}{dx} (1 - x^2) = -2x.$

Now apply L'Hôpital's Rule: $\ln L = \lim_{x \to -1} \frac{\ln (x^2)}{1 - x^2} = \lim_{x \to -1} \frac{\frac{2}{x}}{-2x}.$

Simplify: $\ln L = \lim_{x \to -1} \frac{2}{-2x^2}.$

As $x \to -1$, $x^2 \to 1$, so: $\ln L = \frac{2}{-2 \cdot 1} = -1.$

Exponentiate to Solve for $L$:

We have $\ln L = -1$, so exponentiate both sides: $L = e^{-1}.$

Thus, the value of the limit is: $\lim_{x \to -1} \left( x^2 \right)^{\frac{1}{1 - x^2}} = \frac{1}{e}.$

Final Answer:

$\boxed{\frac{1}{e}}$

Let me know if you would like any further clarifications or additional steps!

Related Questions:

1. How do you handle $1^\infty$ indeterminate forms using logarithms?
2. What are the steps for applying L'Hôpital's Rule in limits?
3. How do you differentiate logarithmic functions like $\ln(x^2)$?
4. Why does exponentiating both sides help solve limits involving logs?
5. What other forms of indeterminate limits exist, and how are they resolved?

Tip: When solving $1^\infty$ forms, always use the logarithm to convert the problem into a solvable fraction!