Evaluate the limit: lim (t-0) x / (3 - sqrt(x+9))

Question

Evaluate the limit: lim (t->0) x / (3 - sqrt(x+9))

ScanSolve AI Tutor · Accepted Answer

The problem asks to evaluate the limit:

\lim_{t \to 0} \frac{x}{3 - \sqrt{x+9}}

There appears to be a variable mismatch, as the limit is for $t \to 0$ but the expression contains $x$ . Assuming the limit should be with respect to $x$ , we will evaluate:

\lim_{x \to 0} \frac{x}{3 - \sqrt{x+9}}

Attempt direct substitution.
Substitute $x=0$ into the expression:
$\frac{0}{3 - \sqrt{0+9}} = \frac{0}{3 - \sqrt{9}} = \frac{0}{3 - 3} = \frac{0}{0}$
This is an indeterminate form, so we need to manipulate the expression.
Rationalize the denominator.
Multiply the numerator and denominator by the conjugate of the denominator, which is $3 + \sqrt{x+9}$ .
$\lim_{x \to 0} \frac{x}{3 - \sqrt{x+9}} \times \frac{3 + \sqrt{x+9}}{3 + \sqrt{x+9}}$
Simplify the expression.
Using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ for the denominator:
$(3 - \sqrt{x+9})(3 + \sqrt{x+9}) = 3^2 - (\sqrt{x+9})^2 = 9 - (x+9) = 9 - x - 9 = -x$
The expression becomes:
$\lim_{x \to 0} \frac{x(3 + \sqrt{x+9})}{-x}$
Cancel common factors.
Since $x \to 0$ , $x$ is not exactly zero, so we can cancel $x$ from the numerator and denominator:
$\lim_{x \to 0} -(3 + \sqrt{x+9})$
Evaluate the limit by direct substitution.
Now substitute $x=0$ into the simplified expression:
$-(3 + \sqrt{0+9})$ $-(3 + \sqrt{9})$ $-(3 + 3)$ $-(6)$ $-6$

Evaluate the limit: lim (t->0) x / (3 - sqrt(x+9))

Answer

Step-by-step solution

Attempt direct substitution.

Rationalize the denominator.

Simplify the expression.

Cancel common factors.

Evaluate the limit by direct substitution.

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