Step 1

Consider the limit

\(\displaystyle\lim_{{{t}\rightarrow{1}}}{\frac{{{t}^{{{4}}}-{t}-{2}}}{{{t}-{1}}}}\)

We want to represent it as a derivative of some function f at some number a, \(\displaystyle{\ln{}}\) light of definition 5 we want to write our limit in the form

\(\displaystyle\lim_{{{t}\rightarrow{a}}}{\frac{{{f{{\left({t}\right)}}}-{f{{\left({a}\right)}}}}}{{{t}-{a}}}}\)

Since \(\displaystyle{t}-{1}\) appears in the denominator, and \(\displaystyle{t}\rightarrow{1}\), we are motivated to take \(\displaystyle{a}={1}\)

Looking at the numerator of the limit in question we want \(\displaystyle{f{{\left({t}\right)}}}={t}^{{{4}}}+{t}\). Let us see if this is the right choice. By definition 5 we have

\(\displaystyle{f}'{\left({1}\right)}=\lim_{{{t}\rightarrow{1}}}{\frac{{{f{{\left({t}\right)}}}-{f{{\left({1}\right)}}}}}{{{t}-{1}}}}\)

\(\displaystyle=\lim_{{{t}\rightarrow{1}}}{\frac{{{t}^{{{4}}}+{t}-{\left({1}^{{{4}}}+{1}\right)}}}{{{t}-{1}}}}\)

\(\displaystyle=\lim_{{{t}\rightarrow{1}}}{\frac{{{t}^{{{4}}}+{t}-{2}}}{{{t}-{1}}}}\)

Which is what we wanted to establish.

Ans: \(\displaystyle{f{{\left({t}\right)}}}={t}^{{{4}}}+{t},\ {a}={1}\)

Consider the limit

\(\displaystyle\lim_{{{t}\rightarrow{1}}}{\frac{{{t}^{{{4}}}-{t}-{2}}}{{{t}-{1}}}}\)

We want to represent it as a derivative of some function f at some number a, \(\displaystyle{\ln{}}\) light of definition 5 we want to write our limit in the form

\(\displaystyle\lim_{{{t}\rightarrow{a}}}{\frac{{{f{{\left({t}\right)}}}-{f{{\left({a}\right)}}}}}{{{t}-{a}}}}\)

Since \(\displaystyle{t}-{1}\) appears in the denominator, and \(\displaystyle{t}\rightarrow{1}\), we are motivated to take \(\displaystyle{a}={1}\)

Looking at the numerator of the limit in question we want \(\displaystyle{f{{\left({t}\right)}}}={t}^{{{4}}}+{t}\). Let us see if this is the right choice. By definition 5 we have

\(\displaystyle{f}'{\left({1}\right)}=\lim_{{{t}\rightarrow{1}}}{\frac{{{f{{\left({t}\right)}}}-{f{{\left({1}\right)}}}}}{{{t}-{1}}}}\)

\(\displaystyle=\lim_{{{t}\rightarrow{1}}}{\frac{{{t}^{{{4}}}+{t}-{\left({1}^{{{4}}}+{1}\right)}}}{{{t}-{1}}}}\)

\(\displaystyle=\lim_{{{t}\rightarrow{1}}}{\frac{{{t}^{{{4}}}+{t}-{2}}}{{{t}-{1}}}}\)

Which is what we wanted to establish.

Ans: \(\displaystyle{f{{\left({t}\right)}}}={t}^{{{4}}}+{t},\ {a}={1}\)