Math Problem Statement
Solution
From the image you uploaded, there are five limits to evaluate. Here are the solutions to each:
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- The exponential function is continuous, and therefore, the limit as approaches any real number is simply the function evaluated at that number. So:
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- The function is a transformation of the function , shifted down by 1. Since the exponential function is continuous, we directly substitute :
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- First, substitute into the function:
- The natural log function is continuous where it is defined (i.e., for ):
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- Since is a continuous function:
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- This can be simplified by recognizing a trigonometric identity:
- Therefore:
Would you like more details or have any further questions about these solutions? Here are some follow-up points to consider:
- What would be the impact on these limits if the approaching values were changed?
- How do these specific functions behave around the limit points?
- Can these solutions be verified using graphical approaches?
- What happens to these functions outside the limits provided?
- How can the concept of limits be used in real-world applications?
Tip: When evaluating limits, especially involving trigonometric functions, remember to check for identities that can simplify calculations.
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Math Problem Analysis
Mathematical Concepts
Limits
Exponential functions
Logarithmic functions
Trigonometric functions
Formulas
lim x→a f(x) = f(a) for continuous functions
lim x→0 sin(x)/x = 1
logarithmic identities for ln(x)
trigonometric identities
Theorems
Limit continuity theorem
Squeeze theorem (for sin and cos limits)
Basic properties of logarithms
Suitable Grade Level
Undergraduate Calculus
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