Use L'Hospital Rule to evaluate the following limits. lim_{xrightarrow0}(tanh x)^x

Use L'Hospital Rule to evaluate the following limits. lim_{xrightarrow0}(tanh x)^x

Question
Limits and continuity
asked 2020-11-30
Use L'Hospital Rule to evaluate the following limits.
\(\lim_{x\rightarrow0}(\tanh x)^x\)

Answers (1)

2020-12-01
Consider the given expression,
\(\lim_{x\rightarrow0}(\tanh x)^x\)
Since the function does not have any denominator hence, the L'Hospital rule cannot be applied.
Thus,
\(\lim_{x\rightarrow0^+}(\tanh x)^x=\lim_{x\rightarrow0^+}(\frac{e^x-e^{-x}}{e^x+e^{-x}})^x\)
\(=(\frac{e^0-e^{-0}}{e^0+e^{-0}})^0\)
\(=1\)
Which is the required value.
0

Relevant Questions

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 2020-10-27
Use L'Hospital Rule to find the limits
\(\lim_{x\rightarrow0}\frac{\sin mx}{\sin nx}\)
asked 2020-12-22
Evaluate the following limit. If you use L'Hospital Rule, be sure to indicate when you are using it, and why it applies.
\(\lim_{x\rightarrow\infty}(3\cdot2^{1-x}+x^2\cdot2^{1-x})\)
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 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 2020-11-26
Use Taylor series to evaluate the following limits.
\(\lim_{x\rightarrow0}\frac{\sqrt{1+2x}-1-x}{x^2}\)
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 2020-10-26
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 2021-02-22
Evaluate the following limits.
\(\lim_{x\rightarrow0}\frac{\tan5x}{x}\)
asked 2021-02-19
Evaluate the following limits.
\(\lim_{x\rightarrow0}\frac{\tan7x}{\sin x}\)
...