Problem 1: Evaluate the limit
$$
\lim_{x \to 0} \frac{(4 + x)^2 - 16}{x}
$$
#### Solution:
1.
Expand the numerator:
$$
(4 + x)^2 = 16 + 8x + x^2
$$
So, the expression becomes:
$$
\frac{(4 + x)^2 - 16}{x} = \frac{16 + 8x + x^2 - 16}{x} = \frac{8x + x^2}{x}
$$
2.
Simplify the fraction:
Factor out \(x\) from the numerator:
$$
\frac{8x + x^2}{x} = \frac{x(8 + x)}{x} = 8 + x \quad \text{(for } x \neq 0\text{)}
$$
3.
Evaluate the limit as \(x \to 0\):
$$
\lim_{x \to 0} (8 + x) = 8
$$
Thus, the answer is:
$$
\boxed{8}
$$
---
Problem 2: Evaluate the limit
$$
\lim_{x \to 0} \frac{5x^3 + 8x^2}{3x^4 - 16x^2}
$$
#### Solution:
1.
Factor out the highest power of \(x\) in both the numerator and the denominator:
- In the numerator, the highest power is \(x^2\):
$$
5x^3 + 8x^2 = x^2(5x + 8)
$$
- In the denominator, the highest power is also \(x^2\):
$$
3x^4 - 16x^2 = x^2(3x^2 - 16)
$$
2.
Simplify the fraction:
$$
\frac{5x^3 + 8x^2}{3x^4 - 16x^2} = \frac{x^2(5x + 8)}{x^2(3x^2 - 16)} = \frac{5x + 8}{3x^2 - 16} \quad \text{(for } x \neq 0\text{)}
$$
3.
Evaluate the limit as \(x \to 0\):
Substitute \(x = 0\) into the simplified expression:
$$
\lim_{x \to 0} \frac{5x + 8}{3x^2 - 16} = \frac{5(0) + 8}{3(0)^2 - 16} = \frac{8}{-16} = -\frac{1}{2}
$$
Thus, the answer is:
$$
\boxed{-\frac{1}{2}}
$$
---
Problem 3: Evaluate the limit
$$
\lim_{x \to 0} \frac{(2 + x)^3 - 8}{x}
$$
#### Solution:
1.
Recognize the structure: The numerator is a difference of cubes. Recall the identity:
$$
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
$$
Here, \(a = 2 + x\) and \(b = 2\). So:
$$
(2 + x)^3 - 8 = (2 + x)^3 - 2^3 = (2 + x - 2)((2 + x)^2 + (2 + x)(2) + 2^2)
$$
Simplify:
$$
(2 + x)^3 - 8 = x((2 + x)^2 + 2(2 + x) + 4)
$$
2.
Substitute back into the limit:
$$
\frac{(2 + x)^3 - 8}{x} = \frac{x((2 + x)^2 + 2(2 + x) + 4)}{x} = (2 + x)^2 + 2(2 + x) + 4 \quad \text{(for } x \neq 0\text{)}
$$
3.
Evaluate the limit as \(x \to 0\):
Substitute \(x = 0\) into the simplified expression:
$$
\lim_{x \to 0} \left((2 + x)^2 + 2(2 + x) + 4\right) = (2 + 0)^2 + 2(2 + 0) + 4 = 2^2 + 2(2) + 4 = 4 + 4 + 4 = 12
$$
Thus, the answer is:
$$
\boxed{12}
$$
---
Final Answers:
1. \(\boxed{8}\)
2. \(\boxed{-\frac{1}{2}}\)
3. \(\boxed{12}\)
Parent Tip: Review the logic above to help your child master the concept of evaluating limits worksheet.
