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Which of the following numbers is equal to \lim_{x \to 0} \dfrac{x^2}{e^{5x}}? a. 2/5 b. 0 c. 5...

Question:

Which of the following numbers is equal to {eq}\lim_{x \to 0} \dfrac{x^2}{e^{5x}} {/eq}?

a. 2/5

b. 0

c. 5

d. None of the above

Limit:

Limit is one of the main concepts of calculus. Differentiation, and integration, concepts of calculus, can be understood easily if we have knowledge of limits. The basic method of limit is the direct substitution method. This method can be written as:

$$\lim_{x\rightarrow a}f(x)=f(a) $$

Answer and Explanation: 1


We have to evaluate the limit:

$$\lim_{x \to 0} \dfrac{x^2}{e^{5x}} $$.

To evaluate the limit of the function, we will apply the direct substitution method.

$$\begin{align} \lim_{x \to 0} \dfrac{x^2}{e^{5x}} &= \dfrac{\lim_{x \to 0} (x^2)}{\lim_{x \to 0} e^{5x}} \ & \left [ \text{Quotient Rule}: \lim_{x \to a} \frac{f(x)}{g(x)}=\frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \right ]\\[0.3cm] &=\dfrac{(0)^2}{e^{5(0)}}\\[0.3cm] &=\dfrac{0}{e^0}\\[0.3cm] &=\dfrac{0}{1}\\[0.3cm] &=0 \end{align} $$

So, the correct option is {eq}\boxed{b.} {/eq}


Learn more about this topic:

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How to Determine the Limits of Functions

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Chapter 6 / Lesson 4
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A limit can tell us the value that a function approaches as that function's inputs get closer and closer to a number. Learn more about how to determine the limits of functions, properties of limits and read examples.


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