Question

How do you prove that the value of limit: ${x^2} - 7x + 3$ is $\left( { - 7} \right)$ as x approaches $2$

Answer 1

In the given question, we are required to find the value of a limit. We can do so by putting the value of the variable as given in the question into the function given to us provided that the limit doesn’t turn into an indeterminate form. We are given a polynomial function in the question and we have to find the limit when the variable x approaches $2$.

Complete step by step solution:
The polynomial function given to us in the question is ${x^2} - 7x + 3$. Hence, the value of limit when x approaches $2$ can be determined by simply putting the value of variable x as $2$ since the limit doesn’t form any indeterminate form.
We represent the given limit in the question as $\mathop {\lim }\limits_{x \to 2} {x^2} - 7x + 3$.
Hence, $\mathop {\lim }\limits_{x \to 2} {x^2} - 7x + 3$
$\Rightarrow {\left( 2 \right)^2} - 7\left( 2 \right) + 3$
$\Rightarrow 4 - 14 + 3$
$\Rightarrow - 7$
Hence, the value of limit $\mathop {\lim }\limits_{x \to 2} {x^2} - 7x + 3$ is $- 7$.

Note: The value of limit that is not of an indeterminate form can be found easily by putting in the value of the variable directly into the function. If the limit is of indeterminate form, then we can employ a variety of methods to convert and solve such limits. There are various types of indeterminate limits like $\dfrac{0}{0}$ form. These kinds of limits can be solved by the L'Hopital's rule easily which involves differentiating the numerator and denominator separately of the rational function in the limit and continuing the limit as it is.