Trigonometric Limits and Theorems Notes - Limits Theorem

Continuity and Principles of Trigonometric Limits

• The sine and cosine functions are characterized by continuity at every real number.

• The tangent, cotangent, secant, and cosecant functions are continuous on their respective domains.

• Verification of trigonometric limits using a calculator requires switching the device to radian mode. This is because the angle measure is based on arc length.

Fundamental Theorems and Corollaries

• Theorem 1:     $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$

• Corollary to Theorem 1:     $\lim_{x \to 0} \frac{x}{\sin(x)} = 1$

• Theorem 2:     $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$

• Useful Trigonometric Identities:

  • Pythagorean Identity adaptation: $1 - \cos^2(x) = \sin^2(x)$

  • Half-angle identity variation: $1 - \cos(x) = 2\sin^2\left(\frac{x}{2}\right)$

Sample Problem 1: Evaluating Limits with Sine

• Problem: Find $\lim_{x \to 0} \frac{\sin(4x)}{x}$

• Solution Procedure:

  • To utilize Theorem 1, the argument of the sine function must match the denominator. Multiply both the numerator and the denominator by $4$.

  • The expression becomes: $4 \times \lim_{x \to 0} \frac{\sin(4x)}{4x}$

  • Because $\lim_{u \to 0} \frac{\sin(u)}{u} = 1$, where $u = 4x$, the limit part evaluates to $1$.

  • Calculation: $4 \times 1 = 4$

  • The required solution is $4$.

Sample Problem 2: Limit involving Reciprocal Sine

• Problem: Find $\lim_{x \to 0} \frac{2x}{\sin(3x)}$

• Solution Procedure:

  • Rewrite the expression to isolate the constant and apply the Corollary to Theorem 1: $\lim_{x \to 0} \frac{x}{\sin(x)} = 1$.

  • Adjustment: $\frac{2}{3} \times \lim_{x \to 0} \frac{3x}{\sin(3x)}$

  • By apply the limit property: $\frac{2}{3} \times 1 = \frac{2}{3}$

  • The required solution is $\frac{2}{3}$.

Sample Problem 3: Limit with Squared Sine and Constant Denominator

• Problem: Find $\lim_{x \to 0} \frac{\sin^2(2x)}{0.5x^2}$

• Solution Procedure:

  • Rewrite the expression to handle the squared function: $\lim_{x \to 0} \frac{1}{0.5} \times \frac{\sin^2(2x)}{x^2}$

  • Recognize that $\frac{1}{0.5} = 2$. Apply the Product Rule for limits: $2 \times \left(\lim_{x \to 0} \frac{\sin(2x)}{x}\right)^2$

  • Evaluate the inner limit by multiplying by $\frac{2}{2}$: $2 \times \left(2 \times \lim_{x \to 0} \frac{\sin(2x)}{2x}\right)^2$

  • Calculation: $2 \times (2 \times 1)^2 = 2 \times 4 = 8$

  • The required solution is $8$.

Sample Problem 4: Limit of Product with Tangent

• Problem: Find $\lim_{t \to 0} \frac{\sin(t) \tan(t)}{t}$

• Solution Procedure:

  • Factor the expression to separate known limits: $\lim_{t \to 0} \left(\frac{\sin(t)}{t} \times \tan(t)\right)$

  • Apply limit properties: $\left(\lim_{t \to 0} \frac{\sin(t)}{t}\right) \times \left(\lim_{t \to 0} \tan(t)\right)$

  • Simplify using Theorem 1 and direct substitution: $1 \times \tan(0)$

  • Calculation: $1 \times 0 = 0$

  • The required solution is $0$.

Sample Problem 5: Applying Theorem 2

• Problem: Find $\lim_{x \to 0} \frac{1 - \cos(2x)}{4x}$

• Solution Procedure:

  • Apply the form found in Theorem 2: $\lim_{x \to 0} \frac{1 - \cos(u)}{u} = 0$.

  • Rewrite the expression: $\frac{1}{2} \times \lim_{x \to 0} \frac{1 - \cos(2x)}{2x}$

  • By applying the Product Rule: $\frac{1}{2} \times 0 = 0$

  • The required solution is $0$.

Sample Problem 6: Using Identities for Squared Cosine Limits

• Problem: Find $\lim_{x \to 0} \frac{1 - \cos^2(\frac{1}{2}x)}{3x^2}$

• Solution Procedure:

  • Use the identity $1 - \cos^2(x) = \sin^2(x)$ to rewrite the numerator as $\sin^2\left(\frac{1}{2}x\right)$.

  • Expression: $\lim_{x \to 0} \frac{\sin^2(\frac{1}{2}x)}{3x^2}$

  • To apply the limit theorem, adjust the denominator to match the square of $\frac{1}{2}x$. Since $\left(\frac{1}{2}x\right)^2 = \frac{1}{4}x^2$, we transform the expression:

  • $\frac{1}{3} \times \lim_{x \to 0} \frac{\sin^2(\frac{1}{2}x)}{x^2}$

  • $\frac{1}{3} \times \lim_{x \to 0} \left(\frac{\sin(\frac{x}{2})}{2 \times \frac{x}{2}}\right)^2$

  • $\frac{1}{3} \times \left(\frac{1}{2}\right)^2 \times \lim_{x \to 0} \left(\frac{\sin(\frac{x}{2})}{\frac{x}{2}}\right)^2$

  • Solve and simplify: $\frac{1}{3} \times \frac{1}{4} \times (1)^2 = \frac{1}{12}$

  • The required solution is $\frac{1}{12}$.

Sample Problem 7: Limit involving Tangent quotient

• Problem: Find $\lim_{x \to 0} \frac{4x}{\tan(x)}$

• Solution Procedure:

  • Use the identity $\tan(x) = \frac{\sin(x)}{\cos(x)}$ to rewrite the limit.

  • $\lim_{x \to 0} \frac{4x}{\frac{\sin(x)}{\cos(x)}} = \lim_{x \to 0} \left(4x \times \frac{\cos(x)}{\sin(x)}\right)$

  • Rearrange the terms: $\lim_{x \to 0} [4 \cos(x)] \times \lim_{x \to 0} [\frac{x}{\sin(x)}]$

  • Since the cosine function is continuous, $\lim_{x \to 0} \cos(x) = \cos(0) = 1$.

  • Using the Corollary to Theorem 1: $[4 \times \cos(0)] \times (1) = 4 \times 1 \times 1 = 4$

  • The required solution is $4$.

Sample Problem 8: Combined Reciprocal Trigonometric Limits

• Problem: Find $\lim_{x \to 0} \frac{x \csc(3x)}{\cot(x)}$

• Solution Procedure:

  • Convert functions using identities: $\csc(3x) = \frac{1}{\sin(3x)}$ and $\cot(x) = \frac{\cos(x)}{\sin(x)}$.

  • Expression: $\lim_{x \to 0} \frac{x \left(\frac{1}{\sin(3x)}\right)}{\frac{\cos(x)}{\sin(x)}} = \lim_{x \to 0} \left(\frac{x}{\sin(3x)} \times \frac{\sin(x)}{\cos(x)}\right)$

  • Split the limit and adjust for Theorem 1 and its Corollary:

  • $\left[\frac{1}{3} \times \lim_{x \to 0} \frac{3x}{\sin(3x)}\right] \times \left[\lim_{x \to 0} \frac{\sin(x)}{x} \times x\right] \times \left[\lim_{x \to 0} \frac{1}{\cos(x)}\right]$

  • Wait, properly simplifying the ratio: $\lim_{x \to 0} \left[\frac{1}{3} \times \frac{3x}{\sin(3x)} \times \frac{\sin(x)}{x} \times \frac{1}{\cos(x)}\right]$

  • Solve and simplify: $\left[\frac{1}{3}\right] \times (1) \times (1) \times \left[\frac{1}{\cos(0)}\right] = \frac{1}{3} \times 1 \times 1 \times 1 = \frac{1}{3}$

  • The required solution is $\frac{1}{3}$.