Question

\[ \lim_{n\rightarrow\infty}\frac{5n^{2}+4}{7n^{2}+6n}= \]

• A. \(0\)
• B. \(\infty\)
• C. \(\dfrac{5}{7}\)
• D. \(\dfrac{7}{5}\)

Hint:

Whenever a limit contains a rational expression in \(n\), first compare the degrees of the numerator and denominator.
• If the degrees are equal, the limit is the ratio of the leading coefficients.
• If the numerator has a smaller degree, the limit is \(0\).
• If the numerator has a larger degree, the limit is \(\infty\) (or \(-\infty\), depending on the sign). Dividing by the highest power of the variable is the quickest and most reliable method.

Answer 1

The Correct Option is C

Concept: The limit of a rational function involving powers of \(n\) can be evaluated by comparing the highest powers of \(n\) in the numerator and denominator. For expressions of the form \[ \lim_{n\to\infty}\frac{a_kn^k+\cdots}{b_kn^k+\cdots}, \] where both numerator and denominator have the same highest degree \(k\), the limit is simply \[ \boxed{\lim_{n\to\infty}\frac{a_kn^k+\cdots}{b_kn^k+\cdots} =\frac{a_k}{b_k}} \] This is obtained by dividing both numerator and denominator by the highest power of \(n\).

Step 1: Identify the highest power of \(n\).
Given, \[ L=\lim_{n\rightarrow\infty}\frac{5n^{2}+4}{7n^{2}+6n}. \] The highest power of \(n\) in both numerator and denominator is \[ n^2. \] Hence, divide both numerator and denominator by \(n^2\). \[ L= \lim_{n\rightarrow\infty} \frac{\dfrac{5n^2}{n^2}+\dfrac{4}{n^2}} {\dfrac{7n^2}{n^2}+\dfrac{6n}{n^2}}. \] Simplifying, \[ L= \lim_{n\rightarrow\infty} \frac{5+\dfrac{4}{n^2}} {7+\dfrac{6}{n}}. \]

Step 2: Evaluate the limit as \(n\rightarrow\infty\).
As \(n\to\infty\), \[ \frac{4}{n^2}\rightarrow0, \qquad \frac{6}{n}\rightarrow0. \] Therefore, \[ L= \frac{5+0}{7+0} = \frac57. \]

Step 3: Write the final answer.
Hence, \[ \boxed{\lim_{n\rightarrow\infty}\frac{5n^{2}+4}{7n^{2}+6n} =\frac57.} \] Therefore, the correct option is \[ \boxed{\text{(C) }\dfrac57.} \]