https://r2.erweima.ai/mathbot/image/1726515145637jgob8zv0.jpg

Let's solve the three limits shown in the image one by one.

### 1. \(\lim_{{x \to 0}} 5x\)

Since \(5x\) is a linear function, we can directly substitute \(x = 0\):

\[
\lim_{{x \to 0}} 5x = 5(0) = 0
\]

### 2. \(\lim_{{x \to -3}} (7x^2 + 4)\)

Here, we are given a quadratic function. Again, we can directly substitute \(x = -3\):

\[
\lim_{{x \to -3}} (7x^2 + 4) = 7(-3)^2 + 4 = 7(9) + 4 = 63 + 4 = 67
\]

### 3. \(\lim_{{x \to -6}} 6x^3\)

This is a cubic function. We can also directly substitute \(x = -6\):

\[
\lim_{{x \to -6}} 6x^3 = 6(-6)^3 = 6(-216) = -1296
\]

### Summary of results:
1. \(\lim_{{x \to 0}} 5x = 0\)
2. \(\lim_{{x \to -3}} (7x^2 + 4) = 67\)
3. \(\lim_{{x \to -6}} 6x^3 = -1296\)

Would you like further details or explanations on these calculations?

Here are 5 related questions to explore further:
1. What is the difference between one-sided and two-sided limits?
2. How do you apply L'Hopital's Rule for indeterminate forms in limits?
3. Can limits involving infinity be solved the same way as finite limits?
4. How do discontinuities affect the computation of limits?
5. What is the formal definition of a limit using epsilon and delta?

**Tip:** When solving limits, always check if direct substitution works before using more advanced techniques like factoring or L'Hopital's Rule.

Evaluate the following limits: 
1. lim(x→0) 5x 
2. lim(x→-3) (7x^2 + 4) 
3. lim(x→-6) 6x^3

Learn how to compute limits of polynomial functions using direct substitution. This problem covers three examples: lim(x→0) 5x, lim(x→-3) (7x^2 + 4), and lim(x→-6) 6x^3, with detailed step-by-step solutions. Suitable for high school students (Grades 10-12).

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