I have opened the image and now can see the four limit problems. Let's solve them step by step:

Problem 1:

$\lim_{{x \to 3^-}} \frac{2}{{x - 3}}$

As $x$ approaches 3 from the left ($3^-$), the denominator $x - 3$ becomes negative, and approaches zero from the negative side. Since the numerator is a positive constant (2), this results in a limit that approaches negative infinity: $\lim_{{x \to 3^-}} \frac{2}{{x - 3}} = -\infty$

Problem 2:

$\lim_{{x \to 3^+}} \frac{2}{{x - 3}}$

Now, $x$ approaches 3 from the right ($3^+$), so $x - 3$ is positive and approaches zero from the positive side. The numerator remains positive, so the result tends towards positive infinity: $\lim_{{x \to 3^+}} \frac{2}{{x - 3}} = \infty$

Problem 3:

$\lim_{{x \to -7}} \frac{1}{{x^2(x + 7)}}$

As $x \to -7$, $(x + 7) \to 0$. In the denominator, $x^2$ approaches a positive value, but $(x + 7)$ approaches zero. So the denominator approaches zero from the negative side, while the numerator is a positive constant (1). Hence, the overall limit tends towards negative infinity: $\lim_{{x \to -7}} \frac{1}{{x^2(x + 7)}} = -\infty$

Problem 4:

$\lim_{{x \to 5}} \frac{2}{{(x - 5)^6}}$

As $x \to 5$, the term $(x - 5)^6$ approaches zero, and since it's raised to an even power (6), it remains positive. So, the limit approaches positive infinity: $\lim_{{x \to 5}} \frac{2}{{(x - 5)^6}} = \infty$

Do you have any questions or would you like more details on any of the solutions?

Here are five related questions you might consider:

1. How do limits behave when approaching from the left versus the right?
2. What is the difference between one-sided and two-sided limits?
3. How does the power of a term affect the behavior of a limit?
4. What happens when a denominator approaches zero while the numerator is a constant?
5. How do limits change when functions involve polynomials of higher degrees?

Tip: When solving limits that involve division by a term approaching zero, always check the sign of the term to determine if the limit tends toward positive or negative infinity.