Mastering Infinite Limits: Techniques For Evaluating Functions Approaching Infinity

Infinite limits

lim ( x→ a) f(x) = ∞. is an infinite limit. This means that as x approaches zero, f(x) gets larger and larger. Examplelim ( x→ 0) 1/x^2 = ∞.

An infinite limit is a special type of limit that occurs when the output of a function approaches positive or negative infinity as the input approaches a specific value or goes towards infinity. In other words, an infinite limit is a limit that does not exist in the traditional sense of having a finite value.

There are three possible ways in which a limit can tend towards infinity:

1. A limit can tend towards positive infinity as the input approaches a specific value or goes towards infinity. This means that the output of the function keeps increasing without upper bound.

2. A limit can tend towards negative infinity as the input approaches a specific value or goes towards infinity. This means that the output of the function keeps decreasing without lower bound.

3. A limit can oscillate, which means that it keeps switching between positive and negative infinity. This can happen when the function bounces back and forth across a horizontal asymptote.

To evaluate an infinite limit, we first need to determine which type of infinity the function approaches. Once we identify the type of infinity, we can use algebraic manipulation to simplify the limit expression and look for patterns. We can also use graphical representations to visualize the behavior of the function as the input approaches infinity.

Some common techniques used to evaluate infinite limits include:

1. Direct substitution: This involves plugging in the infinity value into the function and seeing what the output is. This only works if the resulting expression is an indeterminate form such as 1/0, for example, which requires further simplification.

2. Factoring: This is useful when the expression involves rational functions with factors that can be cancelled out to simplify the expression.

3. L’Hopital’s rule: This rule can be used to evaluate limits that are in an indeterminate form of 0/0 or infinity/infinity. It involves taking the derivative of the numerator and denominator separately until a finite limit is obtained.

4. Comparison theorem: This involves comparing the given function with a simpler function whose limit is already known. This can be used to determine the behavior of the given function as the input approaches infinity.

It is important to note that if a limit tends towards positive or negative infinity, it does not mean that the function is continuous at that point. A function can still be discontinuous even if its limit approaches infinity.

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