Let's break down and solve the limits from the image you've uploaded:

1. $\lim_{x \to 3^+} \log_4(x + 7x - 21)$

First, simplify the expression inside the logarithm: $x + 7x - 21 = 8x - 21$ Now, the limit becomes: $\lim_{x \to 3^+} \log_4(8x - 21)$

Substitute $x = 3$: $8(3) - 21 = 24 - 21 = 3$

Thus, we have: $\log_4(3)$ This is the value of the limit, but since it's a logarithmic expression, it's best expressed as $\log_4(3)$ or you can convert it to another base if needed. Since the base is 4, it's the exact result.

2. $\lim_{t \to \infty} 2t^2 + t + 9$

As $t \to \infty$, the term with the highest degree, $2t^2$, dominates the behavior of the function. The other terms become insignificant as $t$ grows larger. So, we focus on the leading term: $\lim_{t \to \infty} (2t^2 + t + 9) = \infty$

Thus, the limit tends to infinity.

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Summary of Results:

1. $\log_4(3)$
2. $\infty$

Would you like more details or further clarification? Here are some related questions that might help:

1. How to evaluate limits of logarithmic functions as $x \to a$?
2. What is the general approach for limits at infinity for polynomials?
3. How to simplify logarithmic expressions before taking the limit?
4. How to find the behavior of rational functions at infinity?
5. How can we convert a logarithmic expression to a different base?

Tip: When taking limits of polynomials or rational functions at infinity, focus on the highest-degree term, as the lower-degree terms become negligible.