Limit of Indeterminate Form and Algebraic limit - Practice Questions & MCQ

Limit of Indeterminate Form and Algebraic limit

If we directly substitute $\mathrm{x}=\mathrm{a}$ in $\mathrm{f}(\mathrm{x})$ while evaluating $\lim _{x \rightarrow a} f(x)$, and will get one of the seven following forms $\frac{0}{0}, \frac{\infty}{\infty}, \infty\infty, 1^{\infty}, 0^0, \infty^0, \infty \times 0$ then it is called indeterminate form.

For example,
(i) $\lim _{x \rightarrow 2} \frac{x^24}{x2}=\frac{0}{0}$ indeterminate form.
(ii) $\lim _{x \rightarrow 0} \frac{\sin x}{x}=\frac{0}{0}$ indeterminate form.
(iii) $\lim _{x \rightarrow \pi / 2}(\tan x)^{\cos x}=\infty^0$ indeterminate form.

Note:
1. $\frac{0}{0}$ form means the numerator and denominator are both tending to 0 (AND NOT exactly 0 )

Eg, $\lim _{x \rightarrow 0^{+}} \frac{[x]}{x}$, when we input the values of x close to $0($ and $\mathrm{x}>0$ ), then the denominator is tending to 0, but numerator values are not tending to 0 BUT numerator is exactly 0. So this is not $\frac{0}{0}$ form.
2. We can convert one indeterminate form into another and vice versa.

We will divide the problems of  finding limits into five categories, which are

Limit of Algebraic function

Trigonometric Limit

Logarithmic limit

Exponential limit

Miscellaneous forms

One by one we will discuss all of these.

1. Limit of Algebraic function

(a) Direct Substitution Method

To find $\lim _{x \rightarrow a} f(x)$, directly substitute the value of the limit of the variable. (i.e. substitute $x=a$ ) in the expression $\mathrm{f}(\mathrm{x})$
If $f(a)$ is finite then $L=f(a)$
If $f(a)$ is undefined then the limit does not exist.
If $\mathrm{f}(\mathrm{a})$ is indeterminate then this method fails.
(i) $\lim _{x \rightarrow 3}(x(x+1))=3(3+1)=12$
(ii) $\lim _{x \rightarrow 1}\left(\frac{x^2+1}{x+100}\right)=\frac{1+1}{1+100}=\frac{2}{101}$
(iii) $\lim _{x \rightarrow1}\left(1+x+x^2+\ldots+x^{10}\right)=1+(1)+(1)^2+\ldots+(1)^{10}=1$