Math Problem Statement
Solution
Let's evaluate the given limits one by one:
-
: This is a polynomial, and since polynomials are continuous, we can directly substitute :
-
: This is another polynomial, so we can substitute :
-
: Since the logarithmic function is continuous for positive values, substitute : So the limit is , which is approximately 1.791.
-
: The cosine function is continuous, so we can directly substitute :
-
: The sine function is continuous, so we can substitute :
-
: The cosine function oscillates between -1 and 1 as , so this limit does not exist.
-
: The tangent function also oscillates as and does not settle on a specific value, so this limit does not exist.
Summary of results:
- does not exist.
- does not exist.
Would you like more details or clarifications? Here are some related questions:
- What is the general approach for evaluating limits of polynomials?
- How do you evaluate limits of logarithmic functions at specific points?
- What is the behavior of trigonometric functions as approaches infinity?
- What happens when the argument of a trigonometric function becomes unbounded?
- 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 .
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Evaluation of Limits Involving Trigonometric and Exponential Functions
Evaluating Limits Involving Infinity and Logarithmic Functions
Solving Limit Problems Involving Polynomials, Logarithms, and Trigonometric Functions
Evaluating Limits: A Step-by-Step Solution to Complex Limit Expressions
Limit Problems: Evaluate Limits Involving Polynomials, Trigonometric Functions, and Inverses