Evaluate the following limit. lim_{xrightarrow0^+}(1+5x)^{2/x}

Evaluate the following limit. lim_{xrightarrow0^+}(1+5x)^{2/x}

Question
Limits and continuity
asked 2020-11-30
Evaluate the following limit.
\(\lim_{x\rightarrow0^+}(1+5x)^{2/x}\)

Answers (1)

2020-12-01
Evaluate limits:
\(\lim_{x\rightarrow0^+}(1+5x)^{2/x}\)
Apply exponent rule:
\(a^x=e^{\ln(a^x)}=e^{x\cdot\ln(a)}\)
\((1+5x)^{\frac{2}{x}}=e^{\frac{2}{x}\ln(1+5x)}\)
Apply the Limit Chain Rule:
\(g(x)=\frac{2}{x}\ln(1+5x),f(u)=e^u\)
\(\lim_{x\rightarrow0^+}g(x)=\lim_{x\rightarrow0^+}\frac{2}{x}\ln(1+5x)\)
\(=2\cdot\lim_{x\rightarrow0^+}(\frac{\ln(1+5x)}{x})\)
\(\lim_{x\rightarrow0^+}(\frac{\ln(1+5x)}{x})=\frac{0}{0}\)
L'Hospital Rule is used in the following cases,
\(\lim_{x\rightarrow a}(\frac{f(x)}{g(x)})=\frac{0}{0}\ OR\ \lim_{x\rightarrow a}(\frac{f(x)}{g(x)})=\frac{\pm\infty}{\pm\infty}\)
where a can be any real number, \(\infty\ or\ -\infty\)
then,
\(\lim_{x\rightarrow a}(\frac{f(x)}{g(x)})= \lim_{x\rightarrow a}(\frac{f'(x)}{g'(x)})\)
\(\lim_{x\rightarrow0^+}\frac{2}{x}\ln(1+5x)=2\cdot\lim_{x\rightarrow0^+}(\frac{\ln(1+5x)}{x})\)
\(=2\cdot\lim_{x\rightarrow0^+}(\frac{\frac{5}{1+5x}}{1})\)
\(=2\cdot(\frac{5}{1+5(0)})\)
\(=2\cdot5\)
\(=10\)
Result: \(\lim_{x\rightarrow0^+}(1+5x)^{2/x}=e^{10}\)
0

Relevant Questions

asked 2021-01-31
Use Taylor series to evaluate the following limits.
\(\lim_{x\rightarrow0}\frac{\sec x-\cos x-x^2}{x^4} \ (Hint: \text{The Maclaurin series for sec x is }1+\frac{x^2}{2}+\frac{5x^4}{24}+\frac{61x^6}{720}+...)\)
asked 2021-01-13
Evaluate the following limit.
\(\lim_{x\rightarrow0}\frac{\sin ax-\tan^{-1}ax}{bx^3}\)
asked 2020-11-26
Use Taylor series to evaluate the following limits.
\(\lim_{x\rightarrow0}\frac{\sqrt{1+2x}-1-x}{x^2}\)
asked 2020-12-05
Use L'Hospital Rule to evaluate the following limits.
\(\lim_{x\rightarrow0}\frac{\tanh^{-1}x}{\tan(\pi x/2)}\)
asked 2020-10-26
Use Taylor's theorem to evaluate the following limits. \(\lim_{x\rightarrow0}\frac{x\sin(x)-x^2}{\cos(x)-1+\frac{x^2}{2}}\)
asked 2021-03-04
Find the following limit:
\(\lim_{x\rightarrow0}\frac{xe^x}{e^{3x}-1}\)
asked 2021-02-25
Evaluate the following limits. \(\lim_{x\rightarrow0^+}x^{x^2}\)
asked 2021-02-23
Use Taylor's theorem to evaluate the following limits. \(\lim_{x\rightarrow0}\frac{3\sin^2(x)+2\sin^4(x)}{3x\tan(x)}\)
asked 2020-11-09
Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s).
\(\lim_{x\rightarrow0}\frac{e^{ax}-1}{x}\)
asked 2020-12-02
Evaluate the limit
\(\lim_{x\rightarrow\infty}\frac{4x^3-2}{3x^4+5x}\)
...