Question

How do you find the limit of \[\dfrac{{{5^t} - {3^t}}}{t}\] as x approaches \[0\] ?

Answer 1

Hint: Here in this question, we need to find the limit for the function. Let we define the given function as \[f(t)\] and we apply the limit to it. The function will get closer and closer to some number. When the x approaches to \[0\] we need to find the limit of a function.

Complete step by step answer:
The idea of a limit is a basis of all calculus. The limit of a function is defined as let \[f(t)\] be a function defined on an interval that contains \[t = a\]. Then we say that \[\mathop {\lim }\limits_{t \to a} f(t) = L\], if for every \[\varepsilon > 0\]there is some number \[\delta > 0\] such that \[\left| {f(t) - L} \right| < \varepsilon \] whenever\[0 < \left| {t - a} \right| < \delta \].
Here we have to find the value of the limit when x is zero. Let we define the given function as \[f(t) = \dfrac{{{5^t} - {3^t}}}{t}\]. Now we are going to apply the limit to the function \[f(t)\]so we have
\[\mathop {\lim }\limits_{t \to 0} f(t) = \mathop {\lim }\limits_{t \to 0} \dfrac{{{5^t} - {3^t}}}{t}\]
In the numerator add and subtract 1, we get
 \[\mathop {\lim }\limits_{t \to 0} f(t) = \mathop {\lim }\limits_{t \to 0} \dfrac{{{5^t} - 1 - {3^t} + 1}}{t}\]
This can be written as
\[\mathop {\lim }\limits_{t \to 0} f(t) = \mathop {\lim }\limits_{t \to 0} \dfrac{{{5^t} - 1 - ({3^t} - 1)}}{t}\]
The denominator is applied to all terms, so this can be written as
\[\mathop {\lim }\limits_{t \to 0} f(t) = \mathop {\lim }\limits_{t \to 0} \dfrac{{{5^t} - 1}}{t} - \dfrac{{({3^t} - 1)}}{t}\]
The limit is applied to both the terms
\[\mathop {\lim }\limits_{t \to 0} f(t) = \mathop {\lim }\limits_{t \to 0} \dfrac{{{5^t} - 1}}{t} - \mathop {\lim }\limits_{t \to 0} \dfrac{{({3^t} - 1)}}{t}\]
By the formula we know that the \[\mathop {\lim }\limits_{h \to 0} \dfrac{{{a^h} - 1}}{h} = \ln \,a\,\,;\,(a > 0)\]
The function \[f(t)\] is an exponential function. As x approaches to \[0\] we have to substitute the t as\[0\].
\[ \Rightarrow \mathop {\lim }\limits_{t \to 0} f(t) = \ln \,5\, - \,\ln \,3\]
By using the property of the logarithmic functions the above inequality is written as
\[ \Rightarrow \mathop {\lim }\limits_{t \to 0} f(t) = \ln \,\dfrac{5}{3}\]
Therefore we have found the limit for the function \[f(t) = \dfrac{{{5^t} - {3^t}}}{t}\]
Hence \[\mathop {\lim }\limits_{t \to 0} \dfrac{{{5^t} - {3^t}}}{t} = \ln \dfrac{5}{3}\]

Note: We have some standard formulas for the limit, we should be aware of it while solving the problems. We got the solution in the form of logarithm so we have to know about the properties of the logarithmic function and mainly we should know about the definition of the limit function.