Math Problem Statement
Solution
Let's evaluate each limit step by step.
(i)
As , the exponent tends towards , since approaches a small negative value, meaning the denominator approaches 0 from the negative side.
Thus, the expression becomes , which tends to 0.
(ii)
To solve this, we rationalize the numerator by multiplying and dividing by the conjugate:
Simplifying further:
Now, as :
Thus, the limit is:
(iii)
First, factor the numerator:
For , cancel the terms:
Now, as :
Thus, the limit is:
(iv)
This limit is in the standard form of the derivative of the sine function. The derivative of at is:
Thus, the limit is:
Do you want further details or have any questions?
Here are five related questions you might explore:
- How do you evaluate limits involving logarithms?
- What is the limit of exponential functions as the base approaches 0?
- How does rationalizing the numerator help in solving limits?
- What is L'Hopital's rule and when can it be applied?
- How do you find the derivative of trigonometric functions?
Tip: Always check if a limit problem involves an indeterminate form (like ), which may suggest factoring, conjugation, or applying L'Hopital's rule for simplification.
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Math Problem Analysis
Mathematical Concepts
Limits
Exponential Functions
Rationalizing
Derivatives
Trigonometric Limits
Formulas
lim x→a (f(x) - f(a))/(x - a) = f'(a) (definition of derivative)
Standard limit lim x→a (sin(x) - sin(a))/(x - a) = cos(a)
Theorems
Limit laws
Derivative of sine function
Suitable Grade Level
Undergraduate Calculus
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