Question:
The value of $\displaystyle \lim_{x \to 3} \frac{x^{5}-3^{5}}{x^{8}-3^{8}}$ is equal to
The value of $\displaystyle \lim_{x \to 3} \frac{x^{5}-3^{5}}{x^{8}-3^{8}}$ is equal to
Updated On: Jun 7, 2024
- $\frac{5}{8}$
- $\frac{5}{64}$
- $\frac{5}{216}$
- $\frac{1}{27}$
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The Correct Option is C
Solution and Explanation
$\lim\limits _{x \rightarrow 3} \frac{x^{5}-3^{5}}{x^{8}-3^{8}} \,\,\,\left(\right.$ form $\left.\frac{0}{0}\right)$
$=\lim\limits _{x \rightarrow 3} \frac{5 x^{4}}{8 x^{7}} \,\,\,$ (L'Hospital's rule)
$=\lim \limits_{x \rightarrow 3} \frac{5}{8 x^{3}}=\frac{5}{8 \times 3^{3}}=\frac{5}{8 \times 27}=\frac{5}{216}$
$=\lim\limits _{x \rightarrow 3} \frac{5 x^{4}}{8 x^{7}} \,\,\,$ (L'Hospital's rule)
$=\lim \limits_{x \rightarrow 3} \frac{5}{8 x^{3}}=\frac{5}{8 \times 3^{3}}=\frac{5}{8 \times 27}=\frac{5}{216}$
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).