# Limits – Part 1

## Limits of the Form $$\underset{x\to a}{\mathop{\lim }}\,{{\left( f(x) \right)}^{g(x)}}$$ – Part 1

Limits of the Form $$\underset{x\to a}{\mathop{\lim }}\,{{\left( f(x) \right)}^{g(x)}}$$– Form: $${{0}^{0}},{{\infty }^{0}}$$:

Let $$L\text{ }=\underset{x\to a}{\mathop{\lim }}\,{{\left( f(x) \right)}^{g(x)}}$$ then

Taking log on both sides

$${{\log }_{e}}L\text{ }={{\log }_{e}}\left[ \underset{x\to a}{\mathop{\lim }}\,{{\left( f(x) \right)}^{g(x)}} \right]$$,

$${{\log }_{e}}L\text{ }=\left[ {{\log }_{e}}\underset{x\to a}{\mathop{\lim }}\,{{\left( f(x) \right)}^{g(x)}} \right]$$,

$${{\log }_{e}}L\text{ }=\left[ \underset{x\to a}{\mathop{\lim }}\,{{\log }_{e}}{{\left( f(x) \right)}^{g(x)}} \right]$$,

$${{\log }_{e}}L\text{ }=\left[ \underset{x\to a}{\mathop{\lim }}\,g(x){{\log }_{e}}\left( f(x) \right) \right]$$,

$$L\text{ }={{e}^{\left[ \underset{x\to a}{\mathop{\lim }}\,g(x){{\log }_{e}}\left( f(x) \right) \right]}}$$.

Example: Evaluate  $$\underset{x\to \infty }{\mathop{\lim }}\,{{x}^{1/x}}$$

Solution:

Given that $$\underset{x\to \infty }{\mathop{\lim }}\,{{x}^{1/x}}$$

Let us consider$$L=\underset{x\to \infty }{\mathop{\lim }}\,{{x}^{1/x}}$$

Taking log on both sides

$${{\log }_{e}}L={{\log }_{e}}\left( \underset{x\to \infty }{\mathop{\lim }}\,{{x}^{1/x}} \right)$$,

$${{\log }_{e}}L=\left( \underset{x\to \infty }{\mathop{\lim }}\,{{\log }_{e}}{{x}^{1/x}} \right)$$,

$${{\log }_{e}}L=\left( \underset{x\to \infty }{\mathop{\lim }}\,\ \frac{1}{x}\times {{\log }_{e}}x \right)$$,

$$L={{e}^{\left( \underset{x\to \infty }{\mathop{\lim }}\,\ \frac{1}{x}\times {{\log }_{e}}x \right)}}$$,

$$L={{e}^{0}}=1$$.