Math Problem Statement

Evaluate the following limits: 1. lim 3x^2 + 2 as x approaches 1 2. lim 5x^2 + 2 as x approaches 3 3. lim log(2x) as x approaches 3 4. lim cos(x) as x approaches 3 5. lim sin(x) as x approaches 0 6. lim cos(x) as x approaches infinity 7. lim tan(x) as x approaches infinity

Solution

Let's evaluate the given limits one by one:

  1. limx1(3x2+2)\lim_{x \to 1} (3x^2 + 2): This is a polynomial, and since polynomials are continuous, we can directly substitute x=1x = 1: 3(1)2+2=3+2=53(1)^2 + 2 = 3 + 2 = 5

  2. limx3(5x2+2)\lim_{x \to 3} (5x^2 + 2): This is another polynomial, so we can substitute x=3x = 3: 5(3)2+2=5(9)+2=45+2=475(3)^2 + 2 = 5(9) + 2 = 45 + 2 = 47

  3. limx3log(2x)\lim_{x \to 3} \log(2x): Since the logarithmic function is continuous for positive values, substitute x=3x = 3: log(2(3))=log(6)\log(2(3)) = \log(6) So the limit is log6\log 6, which is approximately 1.791.

  4. limx3cos(x)\lim_{x \to 3} \cos(x): The cosine function is continuous, so we can directly substitute x=3x = 3: cos(3)0.9900\cos(3) \approx 0.9900

  5. limx0sin(x)\lim_{x \to 0} \sin(x): The sine function is continuous, so we can substitute x=0x = 0: sin(0)=0\sin(0) = 0

  6. limxcos(x)\lim_{x \to \infty} \cos(x): The cosine function oscillates between -1 and 1 as xx \to \infty, so this limit does not exist.

  7. limxtan(x)\lim_{x \to \infty} \tan(x): The tangent function also oscillates as xx \to \infty and does not settle on a specific value, so this limit does not exist.

Summary of results:

  1. limx1(3x2+2)=5\lim_{x \to 1} (3x^2 + 2) = 5
  2. limx3(5x2+2)=47\lim_{x \to 3} (5x^2 + 2) = 47
  3. limx3log(2x)=log(6)1.791\lim_{x \to 3} \log(2x) = \log(6) \approx 1.791
  4. limx3cos(x)0.9900\lim_{x \to 3} \cos(x) \approx 0.9900
  5. limx0sin(x)=0\lim_{x \to 0} \sin(x) = 0
  6. limxcos(x)\lim_{x \to \infty} \cos(x) does not exist.
  7. limxtan(x)\lim_{x \to \infty} \tan(x) does not exist.

Would you like more details or clarifications? Here are some related questions:

  1. What is the general approach for evaluating limits of polynomials?
  2. How do you evaluate limits of logarithmic functions at specific points?
  3. What is the behavior of trigonometric functions as xx approaches infinity?
  4. What happens when the argument of a trigonometric function becomes unbounded?
  5. How can you determine if a limit of a function exists or does not exist?

Tip: For limits involving trigonometric functions, keep in mind their periodic nature when considering limits as xx \to \infty.

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Math Problem Analysis

Mathematical Concepts

Polynomials
Trigonometric Functions
Logarithmic Functions
Limits

Formulas

lim x→a f(x) = f(a) (if f(x) is continuous at x=a)
lim x→∞ cos(x) = undefined
lim x→∞ tan(x) = undefined

Theorems

Continuity of polynomials
Logarithmic limit properties
Oscillatory behavior of trigonometric functions at infinity

Suitable Grade Level

Grades 11-12