Question

How do you evaluate each of the following limits, if it exists $\displaystyle \lim_{{}}\dfrac{3{{x}^{2}}+5x-2}{{{x}^{2}}-3x-10}$ as $x \to -2$ ?

Answer 1

To evaluate the following given problem, firstly we have to substitute the given limit of $x$ in the given polynomial equation i.e., $\dfrac{3{{x}^{2}}+5x-2}{{{x}^{2}}-3x-10}$ , then we have to compute the polynomial expression using the value of $x$, if the solution found is not in the definite form then we have to apply L-Hospital’s rule for the solution.

Complete step by step solution:
Firstly, we have to substitute the limit of $x$ in the given polynomial equation.
$\displaystyle \lim_{x \to -2}\dfrac{3{{x}^{2}}+5x-2}{{{x}^{2}}-3x-10}$
Apply the limit value,
$\Rightarrow$ $\dfrac{3{{(-2)}^{2}}+5(-2)-2}{{{(-2)}^{2}}-3(-2)-10}$
Now simplify the squares as below,
$\Rightarrow$ $\dfrac{3\times 4+(-10)-2}{4-(-6)-10}$
Open the terms in the bracket,
$\Rightarrow$ $\dfrac{12-10-2}{4+6-10}$
$\Rightarrow$ $\dfrac{12-12}{10-10}$
$\Rightarrow$ $\dfrac{0}{0}$, which is indeterminate (not in the definite form).

Here, the fraction which is in the form $\dfrac{p}{q}$, where $q$ is not equal to $0$ is in the definite form.
Here, we use L-Hospital’s rule in which the function is differentiated and the limit is applied to that,
i.e.
$\displaystyle \lim_{x \to c}\dfrac{f(x)}{g(x)}=\displaystyle \lim_{x \to c}\dfrac{f'(x)}{g'(x)}$
Let us see applying this to the given problem,
$\displaystyle \lim_{x \to -2}\dfrac{3{{x}^{2}}+5x-2}{{{x}^{2}}-3x-10}$
After differentiating the function, we get,
$\Rightarrow$ $\displaystyle \lim_{x \to -2}\dfrac{6x+5}{2x-3}$
Then applying the value of $x$ in the differentiated function,
$\Rightarrow$ $\displaystyle \lim_{x \to -2}\dfrac{6x+5}{2x-3}$ $=\dfrac{6\times (-2)+5}{2(-2)-3}$
$\Rightarrow$ $\dfrac{-12+5}{-4-3}$
$\Rightarrow$ $\dfrac{-7}{-7}$
$\Rightarrow 1$
This is the perfect solution for the given problem.

Note: We have to simplify or evaluate the given problem by applying L-Hospital’s rule. Student can find the difficulty in solving the exponent of negative value and assigning the sign to the negative value, one should pay attention while computing the exponent of negative values, student can also find the difficulty in differentiating and computing the polynomial expressions and one should carefully determine if it is in definite or indefinite form or not.