- #1
NATURE.M
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Homework Statement
a. lim
x→0
(x^3 − 2x + 7)/(3x^2 − 3)
b. lim
x→-1
1/(x+1)
Homework Equations
The Attempt at a Solution
For a. I obtained a limit of 7/-3
For b. the limit does not exist
(I really unsure about this)
Zondrina said:They both look good. Could you elaborate on part (b) a little more though?
NATURE.M said:Thanks.
For b:
lim 1/(x+1) = ∞ (+)
x→-1 (+)
lim 1/(x+1)= ∞ (-)
x→-1 (-)
Therefore, lim 1/(x+1)=undefined
x→-1
A limit is the value that a function approaches as the input variable gets closer and closer to a specific value.
To evaluate a limit using algebraic methods, you can try to simplify the function, factor, or use algebraic properties such as the product rule, quotient rule, or power rule.
In some cases, substitution can be used to evaluate a limit. However, this method only works if the limit is an indeterminate form (such as 0/0 or ∞/∞) and the function is continuous at the specific value.
The Squeeze Theorem, also known as the Sandwich Theorem, states that if two functions sandwich a third function and have the same limit at a specific value, then the third function also has the same limit at that value. This theorem is useful for evaluating limits when the function is complex or undefined at a specific value.
There are three types of limits: one-sided limits, infinite limits, and limits at infinity. One-sided limits are evaluated by approaching the specific value from the left or right side. Infinite limits are evaluated by taking the limit as the input variable approaches positive or negative infinity. Limits at infinity are evaluated by taking the limit as the input variable approaches a specific value, such as 0 or a constant.