Question

consider the sequence defined by ( a_n=\frac{7n - 3}{4n + 1} ).
write down the limit of this sequence.
give an exact answer either as a decimal or as a fraction (in the form ( n/m )) in fully simplified form.

Explanation:

Step1: Divide numerator and denominator by \(n\)

$$\lim_{n ightarrow\infty}a_{n}=\lim_{n ightarrow\infty}\frac{7n - 3}{4n+1}=\lim_{n ightarrow\infty}\frac{7-\frac{3}{n}}{4+\frac{1}{n}}$$

Step2: Use the limit property \(\lim_{n

ightarrow\infty}\frac{1}{n}=0\)
As \(n
ightarrow\infty\), \(\lim_{n
ightarrow\infty}\frac{3}{n}=0\) and \(\lim_{n
ightarrow\infty}\frac{1}{n}=0\). So, \(\lim_{n
ightarrow\infty}\frac{7-\frac{3}{n}}{4+\frac{1}{n}}=\frac{7 - 0}{4+0}\)

Answer:

\(\frac{7}{4}\)