Calculus I: Limits and Limit Laws – Step-by-Step Study Guidance

Q1. Concepts: Left-Hand and Right-Hand Limits, and Limit Theorems

Background

Topic: One-sided limits and limit theorems

This section introduces the concepts of left-hand and right-hand limits, and the conditions under which a two-sided limit exists. It also presents a key trigonometric limit theorem.

Key Terms and Formulas

• Left-Hand Limit: $\lim_{x \to c^-} f(x)$ (as $x$ approaches $c$ from the left)

• Right-Hand Limit: $\lim_{x \to c^+} f(x)$ (as $x$ approaches $c$ from the right)

• Two-Sided Limit: $\lim_{x \to c} f(x) = L$ if and only if $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$

• Special Trig Limit: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

Step-by-Step Guidance

1. Understand that the left-hand limit considers values of $x$ less than $c$, while the right-hand limit considers values greater than $c$.

2. Recall that for the two-sided limit to exist, both one-sided limits must exist and be equal.

3. Recognize the special trigonometric limit and its generalization for expressions approaching zero.

Try summarizing these concepts in your own words before moving on!

Final Answer:

These are foundational concepts for evaluating limits, especially when dealing with piecewise functions or trigonometric expressions. Mastery of one-sided limits and the key trigonometric limit is essential for success in Calculus I.

Q2. Which of the following statements about the function $y = f(x)$ graphed here are true, and which are false?

Background

Topic: Interpreting graphs to determine the truth of statements about limits and function values.

This question tests your ability to read a graph and determine the validity of statements about limits, continuity, and function values.

Key Terms and Formulas

• Limit from a graph: Observe the $y$-value that $f(x)$ approaches as $x$ approaches a specific value from the left and right.

• Continuity: A function is continuous at $x = c$ if $\lim_{x \to c} f(x) = f(c)$.

Step-by-Step Guidance

1. Carefully examine the graph at the points in question. Look for open and closed circles, jumps, or holes.

2. For each statement, identify whether it refers to a limit, a function value, or continuity.

3. Check the left-hand and right-hand limits at the specified $x$-values to see if they agree.

4. Compare the limit (if it exists) to the actual function value at that point.

Try to decide which statements are true or false before checking the answers!

Final Answer:

The truth or falsity of each statement depends on the specific features of the graph at the points in question. Review the graph and apply the definitions of limits and continuity to each statement.

Q3. Find $\lim_{x \to -1^-} f(x)$, $\lim_{x \to -1^+} f(x)$, and $\lim_{x \to -1} f(x)$ where:

$f(x) = \begin{cases} x - 1 & \text{if } x < -1 \\ 0 & \text{if } x = -1 \\ x + 1 & \text{if } x > -1 \end{cases}$

Background

Topic: Evaluating one-sided and two-sided limits for a piecewise function.

This question tests your ability to compute left-hand and right-hand limits for a function defined by different expressions on different intervals.

Key Terms and Formulas

• Piecewise function: A function defined by different formulas for different parts of its domain.

• One-sided limits: $\lim_{x \to c^-} f(x)$ and $\lim_{x \to c^+} f(x)$

• Two-sided limit: $\lim_{x \to c} f(x)$ exists if and only if both one-sided limits exist and are equal.

Step-by-Step Guidance

1. For $x \to -1^-$, use the expression for $x < -1$: $f(x) = x - 1$.

2. For $x \to -1^+$, use the expression for $x > -1$: $f(x) = x + 1$.

3. Evaluate the limit of each expression as $x$ approaches $-1$ from the appropriate side.

4. Compare the left-hand and right-hand limits to determine if the two-sided limit exists.

5. Remember that $f(-1) = 0$, but the value of the function at $x = -1$ does not affect the limit.

Try solving on your own before revealing the answer!

Final Answer:

$\lim_{x \to -1^-} f(x) = -2$

$\lim_{x \to -1^+} f(x) = 0$

$\lim_{x \to -1} f(x)$ does not exist because the left and right limits are not equal.

Q4. Find the following limits:

Background

Topic: Evaluating limits involving radicals, absolute values, and difference quotients.

These problems test your ability to handle limits with square roots, one-sided limits, and expressions that may require algebraic manipulation or special limit laws.

Key Terms and Formulas

• Radical expressions: Expressions involving square roots.

• One-sided limits: Limits as $x$ approaches a value from one side only.

• Difference quotient: Often appears in the form $\frac{f(x+h) - f(x)}{h}$ as $h \to 0$.

• Special limits: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

Step-by-Step Guidance

1. For each limit, first check if direct substitution gives an indeterminate form (like $0/0$).

2. If so, try to simplify the expression. For radicals, consider multiplying numerator and denominator by the conjugate.

3. For absolute values, consider the sign of the expression inside as you approach from the specified side.

4. For difference quotients, look for ways to factor or rationalize to simplify the limit.

5. Set up the simplified expression, but stop before plugging in the final value.

Try solving on your own before revealing the answer!

Final Answer:

(a) $\lim_{x \to 1^+} \frac{\sqrt{x-1}}{x+2} = 0$

(b) $\lim_{h \to 0^+} \frac{\sqrt{h^2 + 4h + 5} - \sqrt{5}}{h} = \frac{2}{\sqrt{5}}$

(c) $\lim_{x \to 1^+} \frac{\sqrt{2x(x-1)}}{|x-1|} = \sqrt{2}$

(d) $\lim_{x \to 1^-} \frac{\sqrt{2x(x-1)}}{|x-1|} = -\sqrt{2}$

Q5. Use $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$ to find the following limits:

Background

Topic: Applying the standard trigonometric limit to more complex expressions.

These problems require you to manipulate the given limit so that it matches the form $\frac{\sin x}{x}$, possibly using substitution or algebraic manipulation.

Key Terms and Formulas

• Standard trig limit: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

• Substitution: Let $u = g(x)$ to rewrite the limit in terms of $u$ as $x \to 0$.

• Algebraic manipulation: Factor, expand, or rewrite expressions to match the standard limit form.

Step-by-Step Guidance

1. For each limit, identify if the numerator or denominator can be rewritten to match the $\frac{\sin x}{x}$ form.

2. If necessary, use substitution (e.g., let $u = \sqrt{2\theta}$ or $u = 2x$) and adjust the limit accordingly.

3. For quotients of sines, use the property $\frac{\sin(a x)}{\sin(b x)}$ as $x \to 0$.

4. For expressions involving $x^2 - x + \sin x$, consider splitting the limit into separate terms.

5. Set up the rewritten limit, but stop before evaluating the final value.

Try solving on your own before revealing the answer!

Final Answer:

(a) $\lim_{\theta \to 0} \frac{\sin \sqrt{2\theta}}{\sqrt{2\theta}} = 1$

(b) $\lim_{x \to 0} \frac{\tan(2x)}{x} = 2$

(c) $\lim_{x \to 0} \frac{x^2 - x + \sin x}{2x} = -\frac{1}{2}$

(d) $\lim_{x \to 0} \frac{\sin(5x)}{\sin(4x)} = \frac{5}{4}$