Лист за преговор: LOGARITHMIC LIMITS AND DERIVATIVES

📋 Course Outline

1. Derivative of Product
2. Primitive of Logarithm
3. Limit of Logarithm at Zero
4. Limit of Logarithm at Infinity
5. Limit of Functions at Zero and Infinity
6. Limit of Logarithm with Function Limit
7. Limit Calculation Techniques
8. Logarithm Limit Examples

📖 1. Derivative of Product

🔑 Key Concepts & Definitions

• Product Rule: If $u(x)$ and $v(x)$ are differentiable functions, then the derivative of their product is: $(uv)' = u'v + uv'$ This rule allows calculating the derivative of a product by differentiating each function separately and combining the results.

• Primitive (Antiderivative): A function $F$ is a primitive of $f$ if $F' = f$.
  In the context of the product rule, the primitive of $\frac{u'}{u}$ (with $u \neq 0$) is $\ln|u|$.

• Logarithmic Derivative: For a differentiable, non-zero function $u$, the derivative of $\ln|u|$ is: $\frac{d}{dx} \ln|u| = \frac{u'}{u}$ This links derivatives of functions to their logarithms, useful in integration and limit calculations.

• Limit Behavior of Logarithms:

  • $\lim_{x \to 0^+} \ln x = -\infty$
  • $\lim_{x \to +\infty} \ln x = +\infty$
    Understanding these helps evaluate limits involving logarithms.

• Limit of $\frac{f(x)}{x}$:

  • If $\lim_{x \to a} f(x) = 0$, then $\lim_{x \to a} \frac{f(x)}{x} = 0$ (under suitable conditions).
    This is useful in analyzing the behavior of functions near specific points.

📝 Essential Points

• The derivative of a product $u(x) v(x)$ is obtained via the product rule: $(uv)' = u'v + uv'$.
• When $u$ is differentiable and non-zero, $\frac{u'}{u}$ is the derivative of $\ln|u|$, which simplifies integration and limit calculations.
• Limits involving $\ln x$ are critical for understanding the behavior of functions near zero and infinity:
  • $\ln x \to -\infty$ as $x \to 0^+$
  • $\ln x \to +\infty$ as $x \to +\infty$
• For functions $U(x)$ with known limits, the limit of $\ln U(x)$ can be deduced based on the limit of $U(x)$:
  • $U(x) \to +\infty \Rightarrow \ln U(x) \to +\infty$
  • $U(x) \to 0^+ \Rightarrow \ln U(x) \to -\infty$
  • $U(x) \to k > 0 \Rightarrow \ln U(x) \to \ln k$

💡 Key Takeaway

The derivative of a product combines the derivatives of each function, and the logarithmic derivative simplifies many calculus operations, especially in limit and integration problems involving products and ratios.

📖 2. Primitive of Logarithm

🔑 Key Concepts & Definitions

• Primitive (Antiderivative): A function $F$ is a primitive of $f$ on an interval $I$ if $F'(x) = f(x)$ for all $x \in I$.

• Logarithmic Primitive: If $u$ is a differentiable function on $I$, not zero, then the primitive of $\frac{u'}{u}$ is $\ln|u|$.

• Chain Rule for Logarithms: $\frac{d}{dx} \ln|u(x)| = \frac{u'(x)}{u(x)}$.

• Limit of Logarithm:

  • $\lim_{x \to 0^+} \ln x = -\infty$
  • $\lim_{x \to +\infty} \ln x = +\infty$

• Behavior of $\ln u(x)$ based on $u(x)$:

  • If $u(x) \to +\infty$, then $\ln|u(x)| \to +\infty$.
  • If $u(x) \to 0^+$, then $\ln|u(x)| \to -\infty$.
  • If $u(x) \to k > 0$, then $\ln|u(x)| \to \ln k$.

📝 Essential Points

• To find an antiderivative involving a logarithm, identify a function $u(x)$ such that $f(x) = \frac{u'(x)}{u(x)}$. Then, the primitive is $F(x) = \ln|u(x)| + C$.

• For limits involving $\ln x$:

  • As $x \to 0^+$, $\ln x \to -\infty$.
  • As $x \to +\infty$, $\ln x \to +\infty$.

• When evaluating limits of $\ln u(x)$, analyze the behavior of $u(x)$:

  • If $u(x) \to 0^+$, the logarithm tends to $-\infty$.
  • If $u(x) \to +\infty$, the logarithm tends to $+\infty$.
  • If $u(x) \to k > 0$, the logarithm tends to $\ln k$.

• For limits involving powers and logarithms:

  • $\lim_{x \to 0^+} x^\alpha \ln x = 0$ for any real $\alpha$.
  • $\lim_{x \to +\infty} \frac{\ln x}{x^\alpha} = 0$.

💡 Key Takeaway

The primitive of $\frac{u'}{u}$ is $\ln|u|$, and understanding the behavior of $\ln u(x)$ near zero or infinity is crucial for evaluating limits involving logarithmic functions.

📖 3. Limit of Logarithm at Zero

🔑 Key Concepts & Definitions

• Limit of a function: The value that a function approaches as the input approaches a specific point.
• Logarithm function ($\ln x$): The inverse of the exponential function $e^x$, defined for $x > 0$.
• Behavior near zero: As $x \to 0^+$, $\ln x \to -\infty$; as $x \to 0^-$, $\ln x$ is undefined.
• Limit involving $\ln x$: When the argument of $\ln x$ approaches zero from the right, the limit tends to $-\infty$.
• Limit involving $\ln x$ at infinity: $\lim_{x \to +\infty} \ln x = +\infty$.
• L'Hôpital's Rule: Used to evaluate indeterminate forms like $0 \cdot (-\infty)$, $\frac{0}{0}$, or $\frac{\infty}{\infty}$.

📝 Essential Points

• Limits at zero:
  • $\lim_{x \to 0^+} \ln x = -\infty$.
  • For functions $f(x)$ with $\lim_{x \to a} U(x) = 0^+$, then $\lim_{x \to a} \ln U(x) = -\infty$.
• Behavior of $\ln x$:
  • $\ln x$ is strictly increasing on $(0, +\infty)$.
  • As $x \to 0^+$, $\ln x$ diverges to $-\infty$.
  • As $x \to +\infty$, $\ln x$ diverges to $+\infty$.
• Limit rules:
  • If $\lim_{x \to a} U(x) = k > 0$, then $\lim_{x \to a} \ln U(x) = \ln k$.
  • For functions involving powers, $\lim_{x \to 0^+} x^\alpha \ln x = 0$ for any real $\alpha$.
• Example calculations:
  • $\lim_{x \to +\infty} \ln(x^n) = +\infty$.
  • $\lim_{x \to 1^+} \ln(x - 1) = -\infty$.
  • $\lim_{x \to -\infty} \ln(x^3 + 3x + 1 / x^3 - 1) = \ln(1) = 0$.

💡 Key Takeaway

The logarithm function tends to $-\infty$ as its argument approaches zero from the right, and understanding this behavior is crucial for evaluating limits involving $\ln x$ near zero and at infinity.

📖 4. Limit of Logarithm at Infinity

🔑 Key Concepts & Definitions

• Limit of a function at infinity: The value a function approaches as the input grows without bound (positive or negative infinity).
  Example: $\lim_{x \to +\infty} f(x)$.

• Logarithm function ($\ln x$): The inverse of the exponential function, defined for $x > 0$.
  Key property: $\ln x \to -\infty$ as $x \to 0^+$; $\ln x \to +\infty$ as $x \to +\infty$.

• Limit involving $\ln x$: As $x \to 0^+$, $\ln x \to -\infty$; as $x \to +\infty$, $\ln x \to +\infty$.
  Note: Limits of the form $\ln x$ are often used to analyze growth rates of functions.

• Behavior of composite functions with $\ln$:

  • If $U(x) \to +\infty$, then $\ln U(x) \to +\infty$.
  • If $U(x) \to 0^+$, then $\ln U(x) \to -\infty$.
  • If $U(x) \to k > 0$, then $\ln U(x) \to \ln k$.

• Limit of the form $\lim_{x \to a} \ln f(x)$:

  • When $f(x) \to +\infty$, $\ln f(x) \to +\infty$.
  • When $f(x) \to 0^+$, $\ln f(x) \to -\infty$.
  • When $f(x) \to k > 0$, $\ln f(x) \to \ln k$.

📝 Essential Points

• Limits involving $\ln x$ are critical in understanding the asymptotic behavior of functions, especially in calculus and analysis.

• Behavior at zero and infinity:

  • $\lim_{x \to 0^+} \ln x = -\infty$.
  • $\lim_{x \to +\infty} \ln x = +\infty$.

• Limit rules for composite functions:

  • If $U(x) \to +\infty$, then $\ln U(x) \to +\infty$.
  • If $U(x) \to 0^+$, then $\ln U(x) \to -\infty$.
  • If $U(x) \to k > 0$, then $\ln U(x) \to \ln k$.

• Common limit results:

  • $\lim_{x \to 0^+} x^\alpha \ln x = 0$ for any real $\alpha$.
  • $\lim_{x \to +\infty} \frac{\ln x}{x^\alpha} = 0$ for any $\alpha > 0$.

• Intuitive technique: Analyze the behavior of the inner function $U(x)$ to determine the limit of $\ln U(x)$.

💡 Key Takeaway

The limit of $\ln x$ at zero and infinity reflects the unbounded growth or decay of logarithmic functions, and understanding the behavior of the inner function $U(x)$ allows for straightforward evaluation of complex limits involving logarithms.

📖 5. Limit of Functions at Zero and Infinity

🔑 Key Concepts & Definitions

• Limit at Zero (x → 0+) or (x → 0−): The value that a function approaches as the variable approaches zero from the right or left.
  Example: limₓ→0+ ln x = -∞

• Limit at Infinity (x → +∞ or x → -∞): The value a function approaches as the variable grows without bound in the positive or negative direction.
  Example: limₓ→+∞ ln x = +∞

• Limit involving logarithms: For functions involving ln(x), the behavior near zero and infinity is critical.
  Key fact: limₓ→0+ ln x = -∞; limₓ→+∞ ln x = +∞

• Limit of a ratio involving functions: If limₓ→a U(x) = L (finite or infinite), then the limit of ln U(x) depends on L:

  • If L = +∞, then limₓ→a ln U(x) = +∞
  • If L = 0+, then limₓ→a ln U(x) = -∞
  • If L = k > 0, then limₓ→a ln U(x) = ln k

• Primitive of a function u: If u is differentiable and non-zero on an interval, then the primitive of u'/u is ln|u|.
  Example: For u(x) = x² + 1, primitive of u'/u = ln|x² + 1|

📝 Essential Points

• Limit of ln(x):

  • As x approaches 0+ from the right, ln x tends to -∞.
  • As x approaches +∞, ln x tends to +∞.

• Behavior of functions involving ln:

  • If a function U(x) tends to +∞ near a point, then ln U(x) also tends to +∞.
  • If U(x) tends to 0+ near a point, then ln U(x) tends to -∞.

• Limit calculations involving ln:

  • When evaluating limₓ→a ln(f(x)), analyze limₓ→a f(x).
  • Use the properties of limits to simplify complex expressions, especially ratios.

• Limit of ratios:

  • For functions like f(x)/x^α, the limit as x approaches 0+ or +∞ often tends to 0 or infinity, depending on α.

• Example techniques:

  • For polynomial ratios, compare degrees.
  • For logarithmic functions, analyze the behavior of the argument.

💡 Key Takeaway

Understanding the behavior of functions involving logarithms near zero and infinity relies on analyzing the limits of their arguments. The limits of ln(x) are fundamental, and the properties of limits of ratios and compositions enable precise evaluation of more complex functions' behavior at these critical points.

📖 6. Limit of Logarithm with Function Limit

🔑 Key Concepts & Definitions

• Primitive (Antiderivative): A function $F$ such that $F' = f$. If $u$ is differentiable and non-zero on an interval, then $\frac{u'}{u}$ has $\ln|u|$ as an antiderivative.

• Limit of Logarithm: Behavior of $\ln x$ as $x$ approaches specific points:

  • $\lim_{x \to 0^+} \ln x = -\infty$
  • $\lim_{x \to +\infty} \ln x = +\infty$

• Limit of a Function Composed with Logarithm: If $U(x) \to L$ as $x \to a$:

  • If $L = +\infty$, then $\lim_{x \to a} \ln U(x) = +\infty$
  • If $L = 0^+$, then $\lim_{x \to a} \ln U(x) = -\infty$
  • If $L = k > 0$, then $\lim_{x \to a} \ln U(x) = \ln k$

• Limit involving powers and logarithms: For any real $\alpha$,

  • $\lim_{x \to 0^+} x^\alpha \ln x = 0$
  • $\lim_{x \to +\infty} \frac{\ln x}{x^\alpha} = 0$

📝 Essential Points

• The logarithm's limit depends on the behavior of its argument:
  • Approaching zero from the right causes $\ln x \to -\infty$.
  • Approaching infinity causes $\ln x \to +\infty$.
• When analyzing limits involving $\ln u(x)$, first determine $\lim_{x \to a} u(x)$:
  • If $u(x) \to +\infty$, then $\ln u(x) \to +\infty$.
  • If $u(x) \to 0^+$, then $\ln u(x) \to -\infty$.
  • If $u(x) \to k > 0$, then $\ln u(x) \to \ln k$.
• For limits involving powers and logs, use dominant term analysis or L'Hôpital's rule when necessary.

💡 Key Takeaway

Understanding the behavior of logarithmic functions near zero and infinity allows for effective evaluation of complex limits, especially when combined with other functions. The key is to analyze the limit of the argument first, then apply the properties of the logarithm accordingly.

📖 7. Limit Calculation Techniques

🔑 Key Concepts & Definitions

• Limit of a function: The value that a function approaches as the input approaches a specific point or infinity. Denoted as $\lim_{x \to a} f(x)$.

• Primitive (antiderivative): A function $F$ such that $F' = f$. For a derivable function $u$ that does not vanish, $\frac{u'}{u}$ has the primitive $\ln|u|$.

• Limit involving $\ln x$:

  • $\lim_{x \to 0^+} \ln x = -\infty$
  • $\lim_{x \to +\infty} \ln x = +\infty$

• Limit of compositions with $\ln$:

  • If $U(x) \to +\infty$, then $\ln U(x) \to +\infty$.
  • If $U(x) \to 0^+$, then $\ln U(x) \to -\infty$.
  • If $U(x) \to k > 0$, then $\ln U(x) \to \ln k$.

• Limit involving powers and logarithms:

  • $\lim_{x \to 0^+} x^\alpha \ln x = 0$ for any real $\alpha$.
  • $\lim_{x \to +\infty} \frac{\ln x}{x^\alpha} = 0$.

📝 Essential Points

• When calculating limits involving $\ln x$, analyze the behavior of the argument:

  • If the argument tends to $+\infty$, the $\ln$ tends to $+\infty$.
  • If the argument tends to 0 from the right, the $\ln$ tends to $-\infty$.
  • If the argument tends to a positive constant $k$, the limit is $\ln k$.

• For limits involving compositions like $\ln U(x)$, first determine the behavior of $U(x)$ as $x \to a$:

  • Use the properties of limits to simplify.
  • Recognize that $\ln$ transforms unbounded growth or decay into $\pm \infty$.

• Use comparison and dominant term analysis for complex expressions:

  • For large $x$, compare numerator and denominator powers.
  • For small $x$, analyze the dominant behavior of the function near zero.

💡 Key Takeaway

Limit calculations involving $\ln x$ rely on understanding the behavior of the argument as $x$ approaches critical points, and applying properties of logarithms and limits to simplify complex expressions. Recognizing the relationship between the growth of functions and their logarithms is essential for accurate evaluation.

📖 8. Logarithm Limit Examples

🔑 Key Concepts & Definitions

• Limit of a logarithm: The value that the logarithmic function approaches as its argument approaches a specific point, often 0, infinity, or a finite value.
• Behavior near zero: As $x \to 0^+$, $\ln x \to -\infty$; as $x \to 0^-$, $\ln x$ is undefined.
• Limit at infinity: As $x \to +\infty$, $\ln x \to +\infty$; growth is slow compared to polynomial functions.
• Limit involving composite functions: If $U(x) \to k > 0$ as $x \to a$, then $\ln U(x) \to \ln k$.
• Limit of the form $x^\alpha \ln x$: For any real $\alpha$, $\lim_{x \to 0^+} x^\alpha \ln x = 0$; similarly, $\lim_{x \to +\infty} x^\alpha \ln x = +\infty$ if $\alpha > 0$.

📝 Essential Points

• Limit rules for $\ln x$:
  • $\lim_{x \to 0^+} \ln x = -\infty$
  • $\lim_{x \to +\infty} \ln x = +\infty$
• Limit of ratios involving $\ln x$:
  • $\lim_{x \to 0^+} \frac{\ln x}{x^\alpha} = 0$ for any $\alpha \in \mathbb{R}$
  • $\lim_{x \to +\infty} \frac{\ln x}{x^\alpha} = 0$ if $\alpha > 0$
• Technique for limits involving $\ln U(x)$:
  • If $U(x) \to +\infty$, then $\ln U(x) \to +\infty$
  • If $U(x) \to 0^+$, then $\ln U(x) \to -\infty$
  • If $U(x) \to k > 0$, then $\ln U(x) \to \ln k$

💡 Key Takeaway

Logarithmic limits depend on the behavior of the argument near critical points; understanding the relationship between $U(x)$ and its limit allows for straightforward evaluation of $\ln U(x)$. When $U(x)$ tends to infinity or zero, the logarithm tends to infinity or negative infinity respectively, simplifying limit calculations.

📊 Synthesis Tables

Topic	Key Concept	Main Formula / Behavior	Typical Use Cases
Derivative of Product	Product Rule	$(uv)' = u'v + uv'$	Differentiating products of functions
Primitive of Logarithm	Primitive of $\frac{u'}{u}$	$\ln	u
Limit of $\ln x$ at Zero	$\lim_{x \to 0^+} \ln x$	$-\infty$	Limits near zero, asymptotic analysis
Limit of $\ln x$ at Infinity	$\lim_{x \to +\infty} \ln x$	$+\infty$	Growth rate comparisons
Limit of Functions at Zero	$\lim_{x \to a} f(x)$	Depends on $f(x)$ behavior	Evaluating limits involving logs and powers
Limit of Logarithm with Function Limit	$\lim_{x \to a} \ln U(x)$	$\ln$ of limit of $U(x)$	Limits involving composition of functions

Topic	Comparison	Key Differences
Limit of $\ln x$ at Zero vs Infinity	Zero: $-\infty$, Infinity: $+\infty$	Behavior at boundary points
Primitive of Logarithm vs Derivative of Product	Primitive: $\ln	u

⚠️ Common Pitfalls & Confusions

1. Confusing $\ln x$ limit at zero with its limit at infinity: $\ln x \to -\infty$ as $x \to 0^+$; $\ln x \to +\infty$ as $x \to +\infty$.

2. Misapplying the product rule: Forgetting to differentiate both functions or mixing terms.

3. Incorrectly handling $\frac{u'}{u}$: Assuming $u$ can be zero; the formula applies only when $u \neq 0$.

4. Neglecting absolute value in $\ln|u|$: Essential when $u$ can be negative; $\ln|u|$ is defined for $u \neq 0$.

5. Misinterpreting limits involving $\ln x$: For example, assuming $\lim_{x \to 0^+} \ln x$ is finite; it diverges to $-\infty$.

6. Forgetting L'Hôpital's rule: Needed when limits are indeterminate forms like $0/0$ or $\infty/\infty$.

7. Assuming $\ln U(x)$ converges when $U(x)$ approaches zero or infinity without analyzing the rate: For example, $\lim_{x \to 0^+} x^\alpha \ln x = 0$ for any $\alpha$, but not vice versa.

8. Ignoring the domain restrictions of $\ln x$: Defined only for $x > 0$; limits approaching zero from the negative side are invalid.

✅ Exam Checklist

• Understand and apply the product rule for derivatives.
• Recognize that the derivative of $\ln|u|$ is $\frac{u'}{u}$ when $u \neq 0$.
• Be able to find primitives of functions involving $\frac{u'}{u}$.
• Know the limits of $\ln x$ as $x \to 0^+$ and $x \to +\infty$.
• Evaluate limits involving $\ln U(x)$ based on the behavior of $U(x)$.
• Use L'Hôpital's rule correctly for indeterminate forms.
• Understand the behavior of functions near zero and infinity, especially involving logarithms.
• Be able to analyze the growth rates of functions involving powers and logarithms.
• Recognize common limit forms and apply appropriate techniques.
• Be familiar with the asymptotic behavior of logarithmic functions.
• Know the domain restrictions of $\ln x$ and avoid invalid limit evaluations.
• Practice evaluating limits involving products, ratios, and compositions with logarithms.
• Confirm the correctness of limit results by analyzing the inner functions' limits.