Question

Find the limit.
$\large{lim_{x→5} \: \frac{ {x}^{2} - 5 }{ {x}^{2} + x - 30} }$


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Answer 1

Answer:

DNE (Doesn’t exist)

Step-by-step explanation:

Hi! We are given the limit expression:

$\displaystyle \large{ \lim_{x \to 5} \frac{x^2-5}{x^2+x-30}}$

First step of evaluating limit is always directly substitution - substitute x = 5 in the expression.

$\displaystyle \large{ \lim_{x\to 5}\frac{5^2-5}{5^2+5-30}}\\ \displaystyle \large{ \lim_{x\to 5}\frac{25-5}{25+5-30}}\\ \displaystyle \large{ \lim_{x\to 5}\frac{20}{0}}$

Looks like we’ve got 20/0 after direct substitution. Note that this isn’t an indeterminate form but undefined since it’s not 0/0 which would make things different.

Now, plot the graph and see at x approaches 5, the function y approaches both positive infinity and negative infinity.

Introducing, left limit would be:

$\displaystyle \large{ \lim_{x \to 5^{-}} \frac{x^2-5}{x^2+x-30} = -\infty}$

And right limit is:

$\displaystyle \large{ \lim_{x \to 5^{+}} \frac{x^2-5}{x^2+x-30} = \infty}$

In limit, if both left and right limit are not equal then the limit does not exist.

From left limit and right, both are not equal. Therefore, the limit does not exist.

Cautions:

• Limit being DNE in this case simply means both left and right limit are not equal.
• The limit does not approach infinity a left limit apporoaches negative infinity as well which is different from positive infinity hence limit being DNE.

Answer 2

$\displaystyle \sf\lim_{x \to5}( \frac{ {x}^{2} - 5 }{ {x}^{2} + x - 30 } )$

Since the function is undefined for 5, evaluate the left-hand and right-hand limits

$\displaystyle \sf\lim_{x \to {5}^{ - } }( \frac{ {x}^{2} - 5 }{ {x}^{2} + x - 30 } ) \\\displaystyle \sf\lim_{x \to {5}^{ + } }( \frac{ {x}^{2} - 5 }{ {x}^{2} + x - 30 } )$

Evaluate the limit

$\sf - \infin \\ + \infin$

Since the left-hand and the right-hand limits are different, the limit does not exist