https://file.aiquickdraw.com/mathbot/file/17272381748718xkspj2a.jpg

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:
\[
\lim_{{x \to 5}} \frac{2}{{(x - 5)^6}} = \infty
\]

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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.

Evaluate the following limits: 1. lim (x -> 3-) 2/(x - 3), 2. lim (x -> 3+) 2/(x - 3), 3. lim (x -> -7) 1/[x^2(x + 7)], 4. lim (x -> 5) 2/[(x - 5)^6]

Learn how to evaluate limits with detailed solutions to problems involving one-sided limits and cases where functions approach infinity. These include expressions like lim (x -> 3-) 2/(x - 3), lim (x -> -7) 1/[x^2(x + 7)], and others. Suitable for students learning undergraduate calculus.

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