7 Indeterminate forms

Indeterminate forms are mathematical expressions that cannot be evaluated directly because they result in an ambiguous or undefined value. These forms arise when attempting to evaluate limits or perform certain operations involving variables or functions. The 7 most common indeterminate forms encountered in calculus and mathematical analysis are:

1.$\frac {\infty}{\infty}$

This form arises when dividing infinity by infinity. It does not provide enough information to determine the value of the expression. It can be encountered when evaluating limits or when trying to find the derivative of a function.

2. $\frac {0}{0}$

This form arises when dividing zero by zero. it is same as above. It can be encountered when evaluating limits or when trying to find the derivative of a function.

3. $\infty – \infty$

This form arises when subtracting infinity from infinity. It is also undefined and requires careful analysis to determine the limit or behavior of the expression.

4. $0^{0}$

This form arises when raising zero power to zero, It is also undefined and requires careful analysis to determine the limit or behavior of the expression.

5.$1^{\infty}$

This form arises when raising 1 to the power of infinity. The value of this expression is indeterminate and depends on the specific context. It often requires techniques such as logarithms or limits to evaluate

6.$\infty ^{0}$

This form arises when raising zero power to infinity, It is also undefined and requires careful analysis to determine the limit or behavior of the expression.

7. $0 \times \infty$

This form arises when multiply zero to infinity, It is also undefined and requires careful analysis to determine the limit or behavior of the expression.

How to evaluate the indeterminate forms

To evaluate indeterminate forms, various mathematical techniques can be employed .Some of them are
L Hospital’s rule
Taylor series expansions
algebraic manipulations
specific limit properties.

These methods help to transform the expression into a determinate form that can be evaluated.

Examples

1.$ \displaystyle \lim_{x \to 0} \frac {sin x}{x}$

When x approaches 0, both the numerator (sin(x)) and denominator (x) tend to 0. This is an example of the indeterminate form 0/0. To evaluate this limit, we can use L’Hôpital’s rule, which states that if the limit of the derivative of the numerator divided by the derivative of the denominator exists, then it is equal to the limit of the original expression. Taking the derivatives, we get

$ \displaystyle \lim_{x \to 0} \frac {sin x}{x} = \displaystyle \lim_{x \to 0} (cos(x)/1) = cos(0)/1 = 1$

2.$\displaystyle \lim_{x \to 0} \frac {x^2 + 3x)}{2x^2 -5x}$

As x approaches infinity, both the numerator and denominator grow without bound, resulting in the indeterminate form $\frac {\infty}{\infty}$. To evaluate this limit, we can divide both the numerator and denominator by the highest power of x, which in this case is $x^2$

$\displaystyle \lim_{x \to 0} \frac {x^2 + 3x)}{2x^2 -5x}=\displaystyle \lim_{x \to 0} \frac {1 + 3/x)}{2 -5/x}=1/2$

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