Question

Find the value of \[\mathop {\lim }\limits_{x \to 0} \;\left[ {\dfrac{{{e^{5x}}\; - {\text{ }}{e^{4x}}}}{x}} \right]\]
A. \[1\]
B. \[2\]
C. \[4\]
D. \[5\]

Answer 1

Hint: In this question, we need to find the values of \[\mathop {\lim }\limits_{x \to 0} \;\left[ {\dfrac{{{e^{5x}}\; - {\text{ }}{e^{4x}}}}{x}} \right]\]. For this, we have to use the concept of limit here to get the desired result. Also, we will use the following formula of limit to get the desired result.

Formula used: We will use the following formulae of limit to solve this question.
 \[\mathop {\lim }\limits_{x \to 0} \dfrac{{{e^{ax}} - 1}}{ax} = 1\]
Where, \[a\] is an integer.

Complete step-by-step solution:
Given that \[\mathop {\lim }\limits_{x \to 0} \;\left[ {\dfrac{{{e^{5x}}\; - {\text{ }}{e^{4x}}}}{x}} \right]\]
Let \[L = \mathop {\lim }\limits_{x \to 0} \;\left[ {\dfrac{{{e^{5x}}\; - {\text{ }}{e^{4x}}}}{x}} \right]\]
Let us find its value.
By adding 1 and -1 to the numerator, we get
\[L = \mathop {\lim }\limits_{x \to 0} \;\left[ {\dfrac{{{e^{5x}}\; - 1 - {\text{ }}{e^{4x}} + 1}}{x}} \right]\]
But we know that \[\mathop {\lim }\limits_{x \to 0} \dfrac{{{e^{ax}} - 1}}{ax} = 1\]
Now, by arranging the terms, we get
\[L = \mathop {\lim }\limits_{x \to 0} \;\left[ {\dfrac{{{e^{5x}}\; - 1 - {\text{ }}\left( {{e^{4x}} - 1} \right)}}{x}} \right]\]
By arranging the terms, we get
\[L = \mathop {\lim }\limits_{x \to 0} \;\left[ {5 \times \dfrac{{{e^{5x}}\; - 1}}{{5x}} - 4 \times \dfrac{{\left( {{e^{4x}} - 1} \right)}}{{4x}}} \right]\]
By separating the both the limits, we get
\[L = 5\mathop {\lim }\limits_{x \to 0} \;\left[ {\dfrac{{{e^{5x}}\; - 1}}{{5x}}} \right] - 4\mathop {\lim }\limits_{x \to 0} \left[ {\dfrac{{\left( {{e^{4x}} - 1} \right)}}{{4x}}} \right]\]
By applying the formula of a limit such as \[\mathop {\lim }\limits_{x \to 0} \dfrac{{{e^{ax}} - 1}}{ax} = 1\] , we get
\[L = 5\left( 1 \right) - 4\left( 1 \right)\]
By simplifying, we get
\[L = 5 - 4\]
\[L = 1\]
Hence, the value of the limit \[\mathop {\lim }\limits_{x \to 0} \;\left[ {\dfrac{{{e^{5x}}\; - {\text{ }}{e^{4x}}}}{x}} \right]\] is 1.

Therefore, the correct option is (A).

Additional information: A limit is defined in mathematics as the value at which a function approaches the outcome for the given input values. Limits describe integrals, derivatives, and continuity in calculus and mathematical modeling. It is used during the analysis phase and always refers to the behavior and attitude of the function at a specific point.

Note: Many students generally make mistakes while rearranging the terms of the numerator of the given limit according to the formula of limit like \[\mathop {\lim }\limits_{x \to 0} \dfrac{{{e^{ax}} - 1}}{ax} = 1\]. They may get confused with the sign and the limit at \[x \to 0\] or \[x \to \infty \]. Thus, the whole result may get wrong.