Question

practice exercises
19–70. evaluating limits find the following limits or state that they do not exist. assume a, b, c, and k are fixed real numbers.

1. $limlimits_{x \to 4} (3x - 7)$
2. $limlimits_{x \to 1} (-2x + 5)$
3. $limlimits_{x \to -9} 5x$
4. $limlimits_{x \to 6} 4$
5. $limlimits_{x \to 1} (2x^3 - 3x^2 + 4x + 5)$
6. $limlimits_{t \to -2} (t^2 + 5t + 7)$
7. $limlimits_{x \to 1} \frac{5x^2 + 6x + 1}{8x - 4}$
8. $limlimits_{t \to 3} \sqrt3{t^2 - 10}$
9. $limlimits_{p \to 2} \frac{3p}{\sqrt{4p + 1} - 1}$
10. $limlimits_{x \to 2} (x^2 - x)^5$
11. $limlimits_{x \to 3} \frac{-5x}{\sqrt{4x - 3}}$
12. $limlimits_{h \to 0} \frac{3}{\sqrt{16 + 3h} + 4}$

Explanation:

Let's solve problem 19: $\lim\limits_{x \to 4} (3x - 7)$

Step 1: Use the limit laws for linear functions

For a linear function $f(x) = mx + b$, the limit as $x$ approaches $a$ is $f(a)$. So we substitute $x = 4$ into the function $3x - 7$.
$3(4) - 7$

Step 2: Calculate the value

First, calculate $3(4) = 12$. Then subtract 7: $12 - 7 = 5$.

Answer:

$\boldsymbol{5}$