Question

Evaluate the limit if it exists.

$\underset{t\to 0}{\lim }\left(\frac{1}{t}-\frac{1}{t^2+t}\right)$

Short answer

The value of limit of the given function $\text{0}.$

Step-by-step solution

Step 1. Given information.

The limit of function is given here,

$\underset{t\to 0}{\lim }\left(\frac{1}{t}-\frac{1}{t^2+t}\right)$

Step 2. Formula used.

Here, we can’t use direct substitution. Combine the fractions in the numerator, using the common denominator and simplify.

Step 3. Simplification.

Given,

$\begin{array}{c}\underset{t\to 0}{\lim }\left(\frac{1}{t}-\frac{1}{t^2+t}\right)=\left(\frac{t^2+t-t}{t(t^2+t)}\right)\\ =\left(\frac{t^2}{t^3+t^2}\right)\\ =\left(\frac{0}{0}\right)\\ =0.\end{array}$

Therefore, our final answer is$0.$