Question

If $\lim _{x \rightarrow-3}\left[3 x^{2}+a x+a-7\right] /\left(x^{2}+2 x-3\right)$, exists, then $a=$
1) 10
2) 15
3) $-15$
4) $-10$

Answer 1

Hint: A function may get close to two distinct limits. There are two scenarios: one in which the variable approaches its limit by values larger than the limit, and the other by values smaller than the limit. Although the right- and left-hand limits are present in this scenario, the limit is not defined.

Complete step by step Solution:
If it is possible to make a function arbitrarily close to $L$ by selecting values that are getting closer and closer to $a$, then the function is said to have a limit LL at aa. Keep in mind that the limit's value is unrelated to the value at $a$.
Following is the notation:
$\lim _{x \rightarrow a} f(x)=L$
Given that
$\lim _{x \rightarrow-3} \dfrac{3 x^{2}+a x+a-7}{x^{2}+2 x-3}$ exist
$x^{2}+2 x-3=(x+3)(x-1)=0 at x=-3$
$\therefore$ The numerator $=0$ for limit to exist
$3(-3)^{2}+a(-3)+a-7=0$
$27-7-3 a+a=0$
$20=2 a$
$a=10$

Hence, the correct option is 1.

Note: Because the idea of a derivative rests on these limits, the theory of limits of functions is the foundation of calculus. How do limits work? With the use of an example, let's clarify what "limit" means. Consider the function $f(x)$. X should have values that are close to the provided constant "a." Then, the relevant set of values is taken by $f(x)$. Assume that the values of $f(x)$ are around some constant when x is close to 'a'. Let's argue that all x values that are sufficiently close to 'a' but not equal to 'a' can be used to make $f(x)$deviate from 'a' by an arbitrarily small amount. As x gets closer to "a," $f(x)$ is then said to approach limit a.