1.5 Determining Limits Using Algebraic Properties

• The limit of a function describes the value that the function approaches
• The algebraic properties of limits allow us to evaluate limits by breaking complex expressions into simpler parts
• What is the value of \lim_{x \to 2} [(x + 3) + (2x)]</latex> using the Sum Rule?

  9
• What is the limit of $x^{2} +$$1$ as $x$ approaches 2?


  5
• The Difference Rule states that $\lim_{x \to c} [f(x) - g(x)] =$$\lim_{x \to c} f(x) - \lim_{x \to c} g(x)$


  True
• The Sum Rule states that \lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)</latex>


  True
• What does the Power Rule state about limits?

  $\lim_{x \to c} [f(x)]^{n} =$$[\lim_{x \to c} f(x)]^{n}$
• Under what condition is direct substitution valid for calculating limits?

  $f(x)$ is continuous
• The notation $\lim_{x \to 3} (x + 2) =$$5$ means that as x</latex> approaches 3, $f(x)$ approaches 5.


  True
• The limit of f(x) = x + 2</latex> as $x$ approaches 3 is 5.


  True
• What is the condition for applying the Quotient Rule?

  $\lim_{x \to c} g(x) \neq 0$
• Applying algebraic properties of limits allows us to simplify and evaluate complex expressions step-by-step.


  True
• What is the limit of \lim_{x \to - 1} \frac{2x + 5}{x^{2} + 1}?


  $\frac{3}{2}$
• Steps to evaluate the limit using algebraic properties:

  1️⃣ Identify the algebraic properties applicable

  2️⃣ Apply the properties to break down the expression

  3️⃣ Evaluate each simplified part

  4️⃣ Combine the results to find the limit
• In direct substitution, the function must be continuous at the point being evaluated
• The product rule for limits states that $\lim_{x \to c} [f(x) \cdot g(x)] =$$\lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$.product
• Factoring the numerator in \lim_{x \to 2} \frac{x^{2} - 4}{x - 2}</latex> results in $(x + 2)(x - 2)$.(x + 2)(x - 2)
• L'Hôpital's Rule states that $\lim_{x \to c} \frac{f(x)}{g(x)} =$$\lim_{x \to c} \frac{f'(x)}{g'(x)}$ if the conditions are met
• Both $f(x)$ and $g(x)$ must be differentiable to apply L'Hôpital's Rule


  True
• When applying L'Hôpital's Rule to $\lim_{x \to 0} \frac{\sin(x)}{x}$, the limit simplifies to 1
• The limit $\lim_{x \to \infty} \frac{x^{2}}{e^{x}}$ simplifies to 0 after applying L'Hôpital's Rule twice


  True
• Canceling common factors in conjugate expressions simplifies the limit


  True
• The limit of a function describes the value it approaches as its input approaches a certain value
• What does the limit of a function describe as its input approaches a certain value?

  The function's behavior
• What is the Sum Rule for limits?

  \lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)</latex>
• The Power Rule for limits states that \lim_{x \to c} [f(x)]^{n} = [\lim_{x \to c} f(x)]^{n}</latex>.


  True
• Using the Quotient Rule, \lim_{x \to 0} \frac{x + 1}{x^{2} + 1}</latex> simplifies to 1
• Direct substitution applied to \lim_{x \to 2} (x^{2} + 3)</latex> yields a limit of 7
• When factoring $\lim_{x \to 2} \frac{x^{2} - 4}{x - 2}$, the common factor to cancel is x - 2
• Factoring and canceling common factors is a useful technique for simplifying complex limit expressions.

  True
• L'Hôpital's Rule states that if $\lim_{x \to c} \frac{f(x)}{g(x)}$ is in indeterminate form, then it is equal to $\lim_{x \to c} \frac{f'(x)}{g'(x)}$ if the latter exists.
• What is the conjugate of $a +$$b$?


  $a - b$
• Trigonometric identities are essential for simplifying expressions involving trigonometric functions in limits.


  True
• What is the reciprocal identity for $\csc x$?


  $\frac{1}{\sin x}$
• What is the double-angle identity for $\cos(2x)$?


  $\cos^{2} x - \sin^{2} x$
• Match the trigonometric identity with its type:

  Tangent Identity ↔️ $\tan x =$$\frac{\sin x}{\cos x}$

  Reciprocal Identity ↔️ $\sec x =$$\frac{1}{\cos x}$

  Double Angle Identity ↔️ $\cos(2x) =$$\cos^{2} x - \sin^{2} x$
• What is the limit of $f(x) =$$x +$$2$ as $x$ approaches 3?


  5
• Match the limit property with its description:

  Sum Rule ↔️ Limits of a sum equal the sum of the limits

  Product Rule ↔️ Limits of a product equal the product of the limits

  Constant Multiple Rule ↔️ Limits of a constant times a function equal the constant times the limit
• The Product Rule states that $\lim_{x \to c} [f(x) \cdot g(x)] =$$\lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$, assuming both limits exist
• The algebraic properties of limits are used to evaluate limits by simplifying complex expressions