https://file.aiquickdraw.com/mathbot/file/1726730674082lro8yroj.pdf

The document contains a set of problems related to limits from a tutorial. I will solve a few of these limits step-by-step:

### 1. (a) \(\lim_{h \to 0} \frac{\sqrt{4+h} - 2}{h}\)

To solve this, multiply both the numerator and denominator by the conjugate:

\[
\frac{\sqrt{4+h} - 2}{h} \cdot \frac{\sqrt{4+h} + 2}{\sqrt{4+h} + 2} = \frac{(4+h) - 4}{h(\sqrt{4+h} + 2)} = \frac{h}{h(\sqrt{4+h} + 2)} = \frac{1}{\sqrt{4+h} + 2}
\]

Now, take the limit as \(h \to 0\):

\[
\lim_{h \to 0} \frac{1}{\sqrt{4+0} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\]

So, the limit is \( \frac{1}{4} \).

### 1. (b) \(\lim_{h \to 0} \frac{h^2 - 2h}{h}\)

Simplify the expression:

\[
\frac{h^2 - 2h}{h} = h - 2
\]

Now, take the limit as \(h \to 0\):

\[
\lim_{h \to 0} (h - 2) = 0 - 2 = -2
\]

So, the limit is \(-2\).

### 1. (c) \(\lim_{h \to 0} \frac{(1-h^2) - (1+2h)}{3-h^2}\)

Simplify the numerator:

\[
(1-h^2) - (1+2h) = 1 - h^2 - 1 - 2h = -h^2 - 2h
\]

The expression becomes:

\[
\lim_{h \to 0} \frac{-h^2 - 2h}{3 - h^2}
\]

As \(h \to 0\), the numerator becomes 0 and the denominator becomes 3:

\[
\lim_{h \to 0} \frac{0}{3} = 0
\]

So, the limit is 0.

### 1. (d) \(\lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}\)

This is the definition of the derivative of \(x^2\) at any point \(x\). Expand the numerator:

\[
(x+h)^2 - x^2 = x^2 + 2xh + h^2 - x^2 = 2xh + h^2
\]

Now, the expression becomes:

\[
\frac{2xh + h^2}{h} = 2x + h
\]

Take the limit as \(h \to 0\):

\[
\lim_{h \to 0} (2x + h) = 2x + 0 = 2x
\]

So, the limit is \(2x\).

Would you like further explanation or solutions for other parts?

Here are some related questions:
1. How do you apply the conjugate method for simplifying limits?
2. What are the key steps to identifying limits involving polynomials?
3. How do you interpret limits that define a derivative?
4. Why does multiplying by the conjugate help in evaluating limits involving square roots?
5. What does it mean if a limit is zero?

**Tip:** When solving limits, always check if a limit represents the derivative of a function at a point. It can greatly simplify the problem.

This page offers detailed solutions to various limit problems, covering fundamental concepts such as the limit definition of derivatives, polynomials, and key techniques like the conjugate method. Suitable for undergraduate-level students.

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