Question
The answer above is NOT correct. Find the following limit. Notes: Enter "DNE" if limit Does Not Exist. \( \lim _{x \rightarrow 0^{+}} e^{\frac{7}{x^{5}}}= \)
Ask by O'Quinn Li. in the United States
Jan 19,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The limit is \( +\infty \).
Solution
Calculate the limit \( \lim_{x \rightarrow 0^{+}} e^{\frac{7}{x^{5}}} \).
Evaluate the limit by following steps:
- step0: Evaluate the limit:
\(\lim _{x\rightarrow 0^{+}}\left(e^{\frac{7}{x^{5}}}\right)\)
- step1: Rewrite the expression:
\(e^{\lim _{x\rightarrow 0^{+}}\left(\frac{7}{x^{5}}\right)}\)
- step2: Calculate:
\(e^{+\infty}\)
- step3: Calculate:
\(+\infty\)
The limit as \( x \) approaches \( 0^{+} \) of \( e^{\frac{7}{x^{5}}} \) is \( +\infty \).
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Simplify this solution Mind Expander
As \( x \) approaches \( 0^{+} \), the expression \( \frac{7}{x^{5}} \) tends toward \( +\infty \) because the denominator \( x^{5} \) becomes very small (and positive). Consequently, \( e^{\frac{7}{x^{5}}} \) also approaches \( e^{+\infty} \), which translates to infinity. Thus, the limit is \( \lim _{x \rightarrow 0^{+}} e^{\frac{7}{x^{5}}} = +\infty \). The final answer is: \(+\infty\)
