# How do you find lim (sqrtx-1)/(root3x-1) as x->1 using l'Hospital's Rule?

Feb 12, 2017

$= \frac{3}{2}$

#### Explanation:

${\lim}_{x \to 1} \frac{\sqrt{x} - 1}{\sqrt{x} - 1}$

If we plug in $x = 1$, we obtain $\frac{0}{0}$, so this is in indeterminate form and we can use L'Hôpital's Rule.

If we then start the L'Hôpital process, we get:

$= {\lim}_{x \to 1} \frac{\frac{1}{2} {x}^{- \frac{1}{2}}}{\frac{1}{3} {x}^{- \frac{2}{3}}}$

At this point we can try $x = 1$ again, and we get:

$= \frac{\frac{1}{2} {\left(1\right)}^{- \frac{1}{2}}}{\frac{1}{3} {\left(1\right)}^{- \frac{2}{3}}} = \frac{3}{2}$
{"ft_lang_label":"__label__en","ft_lang_prob":0.61605406,"math_prob":1.0000076,"size":326,"snap":"2023-14-2023-23","text_gpt3_token_len":95,"char_repetition_ratio":0.111801244,"word_repetition_ratio":0.0,"special_character_ratio":0.2760736,"punctuation_ratio":0.07692308,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999937,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-05-30T11:49:44Z\",\"WARC-Record-ID\":\"<urn:uuid:e9cf185b-8389-4b6a-888b-4364ed4fa1ff>\",\"Content-Length\":\"33429\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b9365665-b16a-4084-96a0-e5bf67c92f09>\",\"WARC-Concurrent-To\":\"<urn:uuid:86949dc0-df11-43cc-8a97-1958fce51748>\",\"WARC-IP-Address\":\"216.239.32.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-find-lim-sqrtx-1-root3x-1-as-x-1-using-l-hospital-s-rule\",\"WARC-Payload-Digest\":\"sha1:HVDL7XGHRHKTJYPLIZEY2Z2TSPO2JGPW\",\"WARC-Block-Digest\":\"sha1:BGBT3ZXA4N52YOE4AG5KGBTRI5PLR2DZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224645595.10_warc_CC-MAIN-20230530095645-20230530125645-00051.warc.gz\"}"}